Most absurd sequence?

[replacinq commas and « 1 »]

Hello SeqFans,

please skip this msg if you’re not in a joke mood...

P = 2, 3, 5, 7, 13, 23, 29, 17, 31, 19, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 127, 223, 137, 139, 149, 227, 157, 229, 163, 233, ...

P was meant to be the lexico-first seq showing only primes (and no duplicate term) where --ahem-- the simultaneous replacement of every comma by a "1" and every "1" by a comma produces a seq of non-primes (no duplicates either)!

Indeed (if my hand didn’t shiver), P will become Q, matching all requirements:

Q = 21315171, 31231291, 713, 1, 913714, 14314715315916, 16717, 1731791831891971, 2712231, 371, 391, 4912271, 5712291, 6312331...

To compare a few terms of P and Q, look here (fixed font recommended):

P = 2,3,5,7,13,23,29,17,31,19,37,41,43,47,53,59,61,67,71,73,79,83,89,97,127,223,137,139,

Q = 21315171,31231291,713,1,913714,14314715315916,16717,1731791831891971,2712231,371,391

I had in mind to create an "absurd" seq, which would be impossible to program on a computer... Now I have my doubts... Ok, if someone with computer skills had the patience to read so far, and the will to check (and extend) this non-sense, I would be delighted, of course!

P.-S.

Just in case: if someone writes the said program, it might be used to compute 9 more seqs -- the ones where the commas are replaced by 0,2,3,4,5,6,7,8,9 respectively – and vice-versa.

__________

[Lars Blomberg]:

Hello Eric!

> "Impossible to program on a computer"

Hmmm...

If you can formulate and carry out an algorithm with your brain, it must be possible to make a program for it, assuming it has a logical, consistent formulation. If it hasn’t then we are in the realms of superstition :-)

As for the "absurd" sequence, I had to assume that the digit sequence 11 in P is not allowed since it results in ‘,,’ in Q.

And further that zeros are not allowed in any numbers of P.

Then I agree with your result:

P = 2, 3, 5, 7, 13, 23, 29, 17, 31, 19, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 127, 223, 137, 139, 149, 227, 157, 229, 163, 233, 167, 173, 179, 181, 239, 241, 251, 257, 191, 263, 193, 197, 269, 199, 271, 277, 281, 283, 293, 313, 331, 337, 317, 347, 349, 353, 359, 367, 373, 379, 383, 389, 397, 419, 421, 431, 433, 439, 443, 449, 457, 461, 463, 467, 479, 487, 491, 499, 521, 523, 541, 547, 557, 563, 569, 571, 577, 587, 593, 599, 613, 617, 619, 631, 641, 643, 647, 653, 659, 661, 673, 677, 683, 691, 727, 733, 719, 739, 743, 751, 757, 761, 769, 773, 787, 797, 821, 823, 827, 829, 839, 853, 857, 859, 863, 877, 881, 883, 887, 919, 929, 937, 941, 947, 953, 967, 977, 971, 983, 991, 997, 1223, 1229, 1237, 2129, 1249, 1259, 2137, 2141, 2143, 2153, 1277, 1279, 2161, 2213, 2179, 2221, 2237, 1283, 1289, 1297, 2239, 1321, 2243, 1327, 2251, 2267, 1361, 2269, 1367, 2273, 1373, 1381, 2281, 2287, 1399, 1423, 2293, 1427, 2297, 1429, 1433, 1439, 2333, 2339, 1447, 1451, 2341, 2347, 1453, 1459, 2351, 2357, 1481, 2377, 1483, 2381, 2383, 1487, 2371, 2389, 1489, 1493, 2393, 1499, 1523, 2399, 1543, 2417, 2423, 2437, 1549, 1553, 2441, 2447, 2459, 1559, 2467, 1567, 1571, 2473, 1579, 2477, 1583, 1597, 1621, 2521, 2531, 2539, 1627, 2543, 1637, 1657, 2549, 1663, 1667, 1669, 2551, 2557, 1693, 1697, 2579, 1699, 2591, 2593, 1721, 2617, 2621, 2633, 1723, 1733, 2647, 1741, 2657, 1747, 1753, 1759, 2659, 1777, 1783, 1787, 1789, 1823, 2663, 1847, 1861, 2671, 2677, 1867, 1871, 2683, 1873, 2687, 2689, 1877, 1879, 1889, 1931, 2693, 1933, 1949, 2699, 1951, 2729, 2713, 2719, 1973, 1979, 2731, 2741, 2749, 1987, 2753, 1993, 2767, 1997, 1999, 2777, 2789, 2791,...

Q = 21315171, 31231291, 713, 1, 913714, 14314715315916, 16717, 1731791831891971, 2712231, 371, 391, 4912271, 5712291, 6312331, 671, 731, 791, 8, 1239124, 125, 12571, 9, 12631, 931, 9712691, 99127, 1277128, 1283129313, 3133, 133713, 7134713491353135913671373137913831389139714, 9142, 143, 14331439144314491457146, 1463146714791487149, 1499152, 1523154, 1547155715631569157, 157715871593159916, 316, 716, 9163, 164, 1643164716531659166, 167316771683169, 1727173317, 917391743175, 1757176, 1769177317871797182, 182318271829183918531857185918631877188, 1883188719, 919291937194, 1947195319671977197, 1983199, 19971, 2231, 2291, 23712, 291, 2491, 25912, 3712, 4, 12, 4312, 531, 2771, 27912, 6, 122, 312, 791222, 122371, 2831, 2891, 297122391, 32, 122431, 3271225, 122671, 36, 122691, 367122731, 3731, 38, 1228, 122871, 3991, 423122931, 427122971, 4291, 4331, 43912333123391, 4471, 45, 1234, 123471, 4531, 4591235, 123571, 48, 123771, 4831238, 123831, 4871237, 123891, 4891, 493123931, 4991, 523123991, 543124, 712423124371, 5491, 5531244, 12447124591, 559124671, 5671, 57, 124731, 579124771, 5831, 5971, 62, 1252, 1253, 125391, 627125431, 6371, 657125491, 6631, 6671, 6691255, 125571, 6931, 697125791, 6991259, 125931, 72, 126, 71262, 126331, 7231, 733126471, 74, 126571, 7471, 7531, 759126591, 7771, 7831, 7871, 7891, 823126631, 8471, 86, 1267, 126771, 8671, 87, 126831, 87312687126891, 8771, 8791, 8891, 93, 126931, 9331, 949126991, 95, 12729127, 3127, 91, 9731, 9791273, 1274, 127491, 987127531, 993127671, 9971, 99912777127891279, 1, 47, ...

It would be possible to allow zeros in P,

for example P = ,1032,101, which

would give Q = 1,0321,0,

Or one may choose not to allow 0 in Q in which case numbers of the form 101, 1001, 10001, are not allowed in P.

Once we clarify this I will be happy to run with the other digits (except 0) as well.

/Lars B

__________

[Jean-Marc Falcoz] :

J’ai fait un programme qui répond presque à tes critères – j’ai simplement rajouté une contrainte (pour que ce soit plus facile à programmer, j’avoue :o))

La contrainte supplémentaire : Toutes les conditions doivent être remplies À CHAQUE ÉTAPE DE LA CONSTRUCTION.

Par exemple : ta suite commence par 2,3,5,7 ... mais à la 4e étape [donc {2,3,5,7}] 2131517 est premier...

Quand on construit ce type de programme, on est vite confronté à des problèmes de définition ...

__________

[Éric] :

> Quand on construit ce type de programme, on est vite confronté à des problèmes de définition ...

... oui, tu as raison -- et je ne suis pas assez clair, pardon ! Voici comment je vois les choses :

P est la suite que nous voulons construire. Q est la suite "fantôme" qui contraint P.

P ne peut contenir que des premiers en un seul exemplaire -- Q que des non-premiers en un seul exemplaire.

On essaie toujours de prolonger P en testant le plus petit premier (absent de P) tel que ce premier ne produise pas de contradiction dans Q.

Les quatre contradictions dans Q sont:

- que Q commence par une virgule -- interdit

- que Q comporte deux virgules consécutives ou plus -- interdit

- que Q(n) soit premier -- interdit

- que Q(n) commence par un zéro -- interdit

Avec ces règles, nous n'avons pas de pb de construction de P. Il y aura toujours un P(n) qui pourra prolonger P sans occasionner de dommage dans Q.

Ta subtile règle supplémentaire qui construit P’ et Q’ peut-elle entrer en contradiction avec ce qui est dit supra ? Franchement, je l'ignore !

__________

[Jean-Marc Falcoz] :

Voici (sauf erreur de ma part), ce qu’on obtient pour P’ et Q’ [avec (n) allant de 1 à 9 (et 0 à la fin) :

P’(1) = 2, 3, 5, 19, 7, 29, 23, 37, 43, 47, 53, 59, 67, 73, 79, 83, 89, 97, 127, 223, 139, 149, 227, 157, 233, 163, 239, 193, 199, 257, 229, 263, 269, 277, 283, 293, 307, 337, 347, 349, 353, 359, 367, 373, 379, 383, 389, 397, 409, 419, 439, 443, 433, 449, 457, 467, 463, 479, 487, 499, 503, 509, 523, 547, 557, 563, 569, 577, 587, 593, 599, 607, 619, 643, 647, 653, 659, 673, 677, 683, 709, 719, 727, 733, 739, 743, 757, 773, 769, 787, 797, 809, 823, 827, 829, 839, 853, 857, 859, 863, 877, 883, 887, 907, 919, 929, 947, 937, 953, 977, 967, 983, 997, 1237, 2003, 1249, 1259, 2029, 1279, 2027, 2039, 2063, 1289, 1297, 2053, 1303, 1327, 2069, 1399, 1423, 2083, 1427, 2087, 2089, 1429, 1447, 1453, 1459, 2099, 1483, 2203, 1489, 1493, 2207, 1543, 2237, 1549, 1553, 2239, 2243, 1559, 2269, 1567, 1579, 2267, 2273, 1583, 1597, 1609, 2293, 1619, 1627, 2287, 1637, 1657, 2297, 1663, 1667, 1669, 2309, 1693, 1697, 2333, 1699, 2339, 1723, 1747, 1753, 1759, 2347, 1777, 1783, 1789, 1847, 1867, 1873, 2357, 1879, 1889, 1933, 1949, 2377, 2383, 1973, 1979, 2389, 1987, 2393, 1993, 2399, 2423, 1999, 2437, 2447, 2459, 2467, 2473, 2477, 2503, 2539, 2543, 2549, 2557, 2579, 2593, 2609, 2633, 2647, 2657, 2659, 2663, 2677, 2683, 2687, 2689, 2693, 2699, 2707, 2719, 2729, 2749, 2753, 2767, 2777, 2789, 2797, 2803, 2819, 2833, 2843, 2837, 2857, 2879, 2887, 2897, 2903, 2909, 2927, 2939, 2953, 2957, 2963, 2969, 2999, 3019, 3023, 3037, 3049, 3067, 3079, 3083, 3089, 3163, 3169, 3187, 3203, 3209, 3229, 3253, 3257, 3259, 3299, 3307, 3319, 3323, 3329, 3343, 3347, 3359, 3373, 3389, 3433, 3407, 3449, 3457, 3463, 3467, 3469, 3499, 3527, 3529, 3533, 3539, 3547, 3557, 3559, 3583, 3593, 3607, 3623,...

Q’(1) = 213151, 9171291231371431471531591671731791831891971, 2712231, 391, 4912271, 5712331, 6312391, 931, 991257122912631269127712831293130713371347134913531359136713731379138313891397140914, 9143914431433144914571467146314791487149915031509152315471557156315691577158715931599160716, 91643164716531659167316771683170917, 9172717331739174317571773176917871797180918231827182918391853185718591863187718831887190719, 9192919471937195319771967198319971, 237120031, 2491, 259120291, 2791202712039120631, 2891, 297120531, 3031, 327120691, 3991, 423120831, 42712087120891, 4291, 4471, 4531, 459120991, 483122031, 4891, 493122071, 543122371, 5491, 55312239122431, 559122691, 5671, 57912267122731, 5831, 5971, 609122931, 6, 91, 627122871, 6371, 657122971, 6631, 6671, 669123091, 6931, 697123331, 699123391, 7231, 7471, 7531, 759123471, 7771, 7831, 7891, 8471, 8671, 873123571, 8791, 8891, 9331, 94912377123831, 9731, 979123891, 987123931, 99312399124231, 9991243712447124591246712473124771250312539125431254912557125791259312609126331264712657126591266312677126831268712689126931269912707127, 91272912749127531276712777127891279712803128, 912833128431283712857128791288712897129031290912927129391295312957129631296912999130, 91302313037130491306713079130831308913, 6313, 6913, 871320313209132291325313257132591329913307133, 91332313329133431334713359133731338913433134071344913457134631346713469134991352713529135331353913547135571355913583135931360713623,...

P’(2) = 3, 5, 7, 11, 13, 17, 19, 29, 31, 37, 41, 43, 53, 47, 59, 61, 67, 71, 73, 79, 83, 89, 97, 101, 103, 107, 109, 113, 131, 137, 139, 149, 151, 157, 163, 167, 173, 179, 181, 191, 193, 197, 199, 233, 239, 251, 257, 263, 269, 277, 281, 293, 307, 311, 313, 317, 337, 331, 347, 349, 353, 359, 367, 373, 379, 383, 389, 397, 401, 409, 419, 421, 431, 433, 439, 443, 449, 457, 461, 463, 467, 479, 487, 491, 499, 503, 509, 521, 557, 541, 563, 547, 569, 571, 577, 587, 593, 599, 601, 607, 613, 617, 619, 631, 641, 643, 647, 653, 659, 661, 673, 677, 683, 691, 701, 709, 719, 733, 739, 751, 743, 761, 757, 769, 773, 787, 797, 809, 811, 821, 829, 839, 853, 857, 859, 863, 877, 881, 883, 887, 907, 911, 919, 929, 937, 941, 947, 953, 967, 971, 977, 983, 991, 997, 1009, 1013, 1019, 1021, 1031, 1033, 1039, 1049, 1051, 1061, 1063, 1069, 1087, 1091, 1093, 1097, 1103, 1109, 1117, 1129, 1151, 1153, 1163, 1171, 1181, 1187, 1193, 1249, 1277, 1291, 1301, 1303, 1307, 1319, 1321, 1361, 1367, 1373, 1381, 1399, 1429, 1433, 1409, 1439, 1447, 1451, 1453, 1459, 1471, 1481, 1483, 1487, 1489, 1493, 1499, 1511, 1531, 1543, 1549, 1553, 1559, 1567, 1571, 1579, 1583, 1597, 1601, 1607, 1609, 1613, 1619, 1621, 1657, 1637, 1663, 1667, 1669, 1693, 1697, 1699, 1709, 1721, 1733, 1741, 1747, 1753, 1759, 1777, 1783, 1787, 1789, 1801, 1811, 1831, 1847, 1861, 1867, 1871, 1873, 1877, 1879, 1889, 1901, 1907, 1913, 1931, 1933, 1949, 1951, 1973, 1979, 1987, 1993, 1997, 1999, 2111, 2129, 2141, 2143, 2153, 2161, 2309, 2333, 2339, 2341, 2351, 2357, 2371, 2377, 2381, 2393, 2399, 2411, 2417, 2437, 2441, 2447, 2459, 2473, 2477, 2531, 2539, 2543, 2549, 2551, 2579,...

Q’(2) = 325272112132172192, 9231237241243253247259261267271273279283289297210121032107210921132131213721392149215121572163216721732179218121912193219721992, 332, 392, 512, 572, 632, 692, 772, 812, 93230723112313231723372331234723492353235923672373237923832389239724012409241924, 124312433243924432449245724612463246724792487249124992503250925, 12557254125632547256925712577258725932599260126072613261726192631264126432647265326592661267326772683269127012709271927332739275127432761275727692773278727972809281128, 128, 928392853285728592863287728812883288729072911291929, 92937294129472953296729712977298329912997210092101321019210, 1210312103321039210492105121061210632106921087210912109321097211032110921117211, 92115121153211632117121181211872119321, 4921, 7721, 9121301213032130721319213, 12136121367213732138121399214, 9214332140921439214472145121453214592147121481214832148721489214932149921511215312154321549215532155921567215712157921583215972160121607216092161321619216, 1216572163721663216672166921693216972169921709217, 12173321741217472175321759217772178321787217892180121811218312184721861218672187121873218772187921889219012190721913219312193321949219512197321979219872199321997219992, 1112, 1, 92, 1412, 1432, 1532, 1612, 3092, 3332, 3392, 3412, 3512, 3572, 3712, 3772, 3812, 3932, 3992, 4112, 4172, 4372, 4412, 4472, 4592, 4732, 4772, 5312, 5392, 5432, 5492, 5512, 579,...

P’(3) = 2, 5, 11, 7, 17, 19, 29, 31, 41, 47, 59, 61, 67, 71, 79, 89, 97, 101, 107, 109, 127, 131, 149, 139, 157, 151, 167, 179, 181, 191, 197, 199, 211, 227, 229, 239, 269, 241, 257, 251, 271, 277, 281, 349, 401, 409, 419, 421, 431, 439, 449, 457, 461, 467, 479, 487, 491, 499, 509, 521, 541, 547, 557, 569, 571, 577, 587, 599, 601, 607, 617, 619, 631, 641, 647, 661, 677, 659, 691, 701, 709, 727, 719, 739, 751, 757, 769, 761, 787, 797, 809, 811, 821, 827, 829, 839, 857, 859, 877, 881, 887, 907, 911, 919, 929, 947, 941, 967, 971, 977, 991, 997, 1009, 1019, 1021, 1031, 1039, 1049, 1051, 1061, 1069, 1087, 1091, 1097, 1117, 1109, 1129, 1151, 1171, 1181, 1187, 1201, 1217, 1229, 1231, 1259, 1249, 1277, 1279, 1289, 1291, 1297, 1321, 1409, 1327, 1439, 1381, 1399, 1427, 1429, 1447, 1451, 1459, 1471, 1481, 1487, 1489, 1499, 1511, 1531, 1549, 1559, 1567, 1571, 1579, 1597, 1601, 1607, 1609, 1619, 1621, 1627, 1657, 1667, 1669, 1697, 1699, 1709, 1721, 1741, 1747, 1759, 1777, 1787, 1789, 1801, 1811, 1831, 1847, 1861, 1867, 1871, 1877, 1879, 1889, 1901, 1907, 1949, 1931, 1951, 1979, 1987, 1999, 1997, 2011, 2017, 2027, 2029, 2039, 2069, 2081, 2087, 2089, 2099, 2111, 2129, 2131, 2141, 2161, 2207, 2179, 2221, 2239, 2251, 2267, 2269, 2281, 2287, 2297, 2351, 2357, 2377, 2381, 2399, 2417, 2411, 2441, 2447, 2459, 2467, 2477, 2521, 2531, 2539, 2551, 2549, 2557, 2579, 2609, 2591, 2617, 2621, 2647, 2657, 2671, 2659, 2677, 2687, 2689, 2699, 2707, 2711, 2719, 2729, 2731, 2741, 2749, 2767, 2777, 2789, 2791, 2797, 2801, 2819, 2851, 2861, 2857, 2879, 2887, 2897, 2909, 2917, 2927, 2939, 2957, 2969, 2971, 2999, 4001, 3119, 4007, 4019, 3121, 4021, 3169, 4027, 3187, 4049, 3209, 3217,...

Q’(3) = 235311373173193293, 1341347359361367371379389397310131073109312731, 1314931, 93157315131673179318131913197319932113227322932, 932693241325732513271327732813, 49340134093419342134, 134, 9344934573461346734793487349134993509352135413547355735693571357735873599360136073617361936, 1364136473661367736593691370137093727371937, 93751375737693761378737973809381138213827382938, 93857385938773881388739073911391939293947394139673971397739913997310093101931021310, 1310, 93104931051310613106931087310913109731117311093112931151311713118131187312013121731229312, 13125931249312773127931289312913129731, 213140931, 27314, 931, 8131, 993142731429314473145131459314713148131487314893149931511315, 1315493155931567315713157931597316013160731609316193162131627316573166731669316973169931709317213174131747317593177731787317893180131811318, 131847318613186731871318773187931889319013190731949319, 1319513197931987319993199732011320173202732029320, 932069320813208732089320993211132129321, 13214132161322073217932221322, 932251322673226932281322873229732, 5132, 5732, 7732, 8132, 993241732411324413244732459324673247732521325, 1325, 93255132549325573257932609325913261732621326473265732671326593267732687326893269932707327113271932729327, 1327413274932767327773278932791327973280132819328513286132857328793288732897329093291732927329, 932957329693297132999340013, 11934007340193, 121340213, 169340273, 187340493, 2093, 217,...

P’(4) = 2, 3, 5, 7, 17, 13, 19, 11, 23, 29, 31, 37, 41, 61, 53, 59, 67, 73, 71, 79, 83, 89, 97, 101, 103, 107, 109, 113, 127, 131, 137, 139, 149, 157, 151, 163, 167, 173, 179, 181, 191, 193, 197, 199, 211, 223, 227, 229, 233, 239, 241, 257, 263, 251, 269, 271, 277, 281, 283, 293, 307, 311, 313, 317, 331, 337, 349, 353, 359, 367, 373, 379, 383, 389, 397, 421, 433, 439, 457, 463, 487, 491, 499, 503, 509, 521, 523, 541, 569, 563, 557, 571, 577, 587, 593, 599, 601, 607, 613, 617, 619, 631, 641, 659, 653, 661, 673, 677, 683, 691, 701, 709, 719, 727, 733, 739, 751, 757, 761, 769, 773, 787, 797, 809, 811, 821, 823, 827, 829, 839, 853, 859, 857, 863, 877, 881, 883, 887, 907, 911, 919, 929, 937, 941, 953, 967, 971, 977, 983, 991, 997, 1009, 1013, 1019, 1021, 1031, 1033, 1039, 1049, 1051, 1061, 1063, 1069, 1087, 1091, 1093, 1097, 1103, 1109, 1117, 1123, 1129, 1151, 1153, 1163, 1171, 1181, 1187, 1193, 1201, 1213, 1217, 1223, 1229, 1231, 1237, 1249, 1259, 1277, 1279, 1283, 1289, 1291, 1297, 1301, 1303, 1307, 1319, 1321, 1327, 1361, 1367, 1373, 1381, 1399, 1427, 1511, 1433, 1439, 1451, 1481, 1487, 1523, 1493, 1549, 1531, 1499, 1553, 1559, 1567, 1571, 1579, 1583, 1597, 1601, 1607, 1609, 1613, 1619, 1621, 1627, 1637, 1657, 1663, 1667, 1669, 1693, 1697, 1699, 1721, 1709, 1723, 1733, 1741, 1753, 1759, 1777, 1783, 1787, 1789, 1801, 1811, 1823, 1831, 1861, 1867, 1871, 1873, 1877, 1879, 1889, 1901, 1907, 1913, 1931, 1933, 1949, 1951, 1973, 1979, 1987, 1993, 1997, 1999, 2003, 2011, 2017, 2027, 2029, 2039, 2053, 2063, 2069, 2081, 2083, 2087, 2089, 2099, 2111, 2113, 2129, 2131, 2137, 2141, 2153, 2161, 2179, 2203, 2207, 2213,...

Q’(4) = 24345474174134194114234294314374, 146145345946747347147948348949741014103410741094113412741314137413941, 94157415141634167417341794181419141934197419942114223422742294233423942, 142574263425142694271427742814283429343074311431343174331433743, 9435343594367437343794383438943974, 214, 334, 394, 574, 634, 874, 914, 99450345094521452345, 14569456345574571457745874593459946014607461346174619463146, 1465946534661467346774683469147014709471947274733473947514757476147694773478747974809481148214823482748294839485348594857486348774881488348874907491149194929493749, 1495349674971497749834991499741009410134101941021410314103341039410, 9410514106141063410694108741091410934109741103411094111741123411294115141153411634117141181411874119341201412134121741223412294123141237412, 941259412774127941283412894129141297413014130341307413194132141327413614136741373413814139941, 274151141, 3341, 3941, 5141, 8141, 874152341, 93415, 94153141, 994155341559415674157141579415834159741601416074160941613416194162141627416374165741663416674166941693416974169941721417094172341733417, 141753417594177741783417874178941801418114182341831418614186741871418734187741879418894190141907419134193141933419, 94195141973419794198741993419974199942003420114201742027420294203942053420634206942081420834208742089420994211142113421294213142137421, 1421534216142179422034220742213,...

P’(5) = 2, 3, 11, 7, 13, 17, 19, 23, 29, 41, 31, 37, 43, 47, 59, 61, 67, 73, 71, 83, 79, 89, 97, 101, 103, 109, 107, 113, 127, 131, 137, 139, 149, 151, 163, 167, 173, 179, 181, 191, 197, 193, 199, 211, 223, 227, 229, 233, 239, 241, 251, 281, 263, 269, 271, 277, 293, 283, 307, 311, 313, 317, 331, 337, 347, 349, 359, 367, 373, 379, 383, 389, 397, 401, 409, 419, 421, 431, 433, 439, 443, 449, 461, 463, 467, 479, 487, 491, 499, 521, 563, 569, 577, 587, 593, 599, 601, 607, 613, 619, 631, 617, 641, 643, 647, 659, 661, 673, 677, 683, 691, 701, 709, 719, 727, 733, 739, 751, 743, 761, 769, 773, 787, 797, 809, 811, 821, 823, 827, 829, 839, 859, 863, 877, 881, 883, 887, 907, 911, 919, 929, 937, 941, 947, 967, 971, 977, 983, 991, 997, 1009, 1013, 1019, 1021, 1031, 1033, 1039, 1049, 1051, 1061, 1063, 1069, 1087, 1091, 1093, 1097, 1103, 1109, 1117, 1123, 1129, 1151, 1181, 1163, 1171, 1187, 1193, 1201, 1213, 1217, 1223, 1229, 1231, 1237, 1249, 1259, 1279, 1277, 1283, 1289, 1291, 1297, 1301, 1303, 1307, 1319, 1321, 1327, 1361, 1367, 1373, 1381, 1399, 1409, 1423, 1427, 1429, 1433, 1439, 1447, 1451, 1459, 1471, 1481, 1483, 1487, 1489, 1493, 1499, 1549, 1601, 1607, 1609, 1613, 1619, 1621, 1627, 1637, 1663, 1667, 1669, 1693, 1697, 1699, 1709, 1721, 1723, 1733, 1741, 1747, 1759, 1777, 1783, 1787, 1789, 1811, 1801, 1823, 1831, 1847, 1861, 1867, 1871, 1873, 1877, 1879, 1889, 1901, 1907, 1913, 1931, 1933, 1949, 1951, 1973, 1979, 1987, 1993, 1997, 1999, 2003, 2011, 2017, 2027, 2029, 2039, 2063, 2069, 2081, 2083, 2087, 2089, 2099, 2111, 2113, 2129, 2131, 2137, 2141, 2143, 2161, 2179, 2203, 2207, 2221, 2213, 2237, 2239, 2243, 2251,...

Q’(5) = 253511575135175195235295415315375435475, 9561567573571583579589597510151035109510751135127513151375139514951, 1516351675173517951815191519751935199521152235227522952335239524152, 152815263526952715277529352835307531153135317533153375347534953, 953675373537953835389539754015409541954215431543354395443544954615463546754795487549154995, 215, 635, 695, 775, 875, 935, 9956015607561356195631561756415643564756, 95661567356775683569157015709571957275733573957, 1574357615769577357875797580958115821582358275829583958, 95863587758815883588759075911591959295937594159475967597159775983599159975100951013510195102151031510335103951049510, 1510615106351069510875109151093510975110351109511175112351129511, 151181511635117151187511935120151213512175122351229512315123751249512, 9512795127751283512895129151297513015130351307513195132151327513615136751373513815139951409514235142751429514335143951447514, 1514, 95147151481514835148751489514935149951, 495160151607516095161351619516215162751637516635166751669516935169751699517095172151723517335174151747517, 951777517835178751789518115180151823518315184751861518675187151873518775187951889519015190751913519315193351949519, 15197351979519875199351997519995200352011520175202752029520395206352069520815208352087520895209952111521135212952131521375214152143521615217952203522075222152213522375223952243522, 1,...

P’(6) = 2, 5, 3, 7, 13, 11, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 71, 73, 79, 83, 89, 97, 101, 103, 107, 109, 113, 127, 131, 137, 139, 149, 151, 157, 173, 179, 181, 191, 193, 197, 199, 211, 223, 227, 229, 233, 239, 241, 251, 257, 269, 271, 277, 281, 293, 283, 307, 311, 313, 317, 331, 337, 347, 349, 353, 359, 373, 379, 383, 389, 401, 397, 409, 419, 421, 431, 433, 439, 443, 449, 457, 461, 479, 487, 491, 499, 503, 509, 523, 521, 541, 547, 557, 569, 571, 577, 587, 593, 599, 677, 691, 701, 709, 719, 727, 733, 739, 743, 751, 757, 761, 773, 769, 809, 787, 797, 811, 821, 823, 827, 829, 839, 853, 857, 859, 877, 881, 883, 887, 907, 911, 919, 929, 937, 941, 947, 953, 971, 977, 983, 991, 997, 1009, 1013, 1019, 1021, 1031, 1033, 1039, 1049, 1051, 1061, 1069, 1093, 1087, 1091, 1097, 1103, 1109, 1117, 1123, 1129, 1151, 1153, 1171, 1181, 1187, 1193, 1201, 1213, 1217, 1223, 1229, 1231, 1237, 1249, 1259, 1277, 1279, 1283, 1289, 1291, 1297, 1301, 1303, 1307, 1319, 1321, 1327, 1361, 1373, 1381, 1399, 1409, 1423, 1427, 1433, 1429, 1439, 1447, 1451, 1453, 1459, 1471, 1481, 1483, 1487, 1489, 1493, 1499, 1511, 1523, 1531, 1543, 1549, 1553, 1559, 1571, 1579, 1583, 1597, 1621, 1709, 1627, 1657, 1693, 1699, 1721, 1723, 1733, 1741, 1747, 1753, 1759, 1777, 1787, 1783, 1801, 1789, 1811, 1823, 1831, 1847, 1861, 1871, 1873, 1877, 1879, 1889, 1901, 1907, 1913, 1931, 1933, 1949, 1951, 1973, 1979, 1987, 1993, 1997, 1999, 2003, 2011, 2017, 2027, 2029, 2039, 2053, 2069, 2081, 2083, 2087, 2089, 2099, 2111, 2113, 2129, 2131, 2137, 2141, 2143, 2153, 2161, 2179, 2203, 2207, 2213, 2221, 2237, 2239, 2243, 2251, 2269, 2273, 2281, 2287, 2293,...

Q’(6) = 26563676136116176196236296316376416436476536596, 1671673679683689697610161036107610961136127613161376139614961516157617361796181619161936197619962116223622762296233623962416251625762, 962716277628162936283630763116313631763316337634763496353635963736379638363896401639764096419642164316433643964436449645764, 16479648764916499650365096523652165416547655765, 9657165776587659365996, 776, 9167016709671967276733673967436751675767, 1677367, 968096787679768116821682368276829683968536857685968776881688368876907691169196929693769416947695369716977698369916997610096101361019610216103161033610396104961051610, 1610, 9610936108761091610976110361109611176112361129611516115361171611816118761193612016121361217612236122961231612376124961259612776127961283612896129161297613016130361307613196132161327613, 16137361381613996140961423614276143361429614396144761451614536145961471614816148361487614896149361499615116152361531615436154961553615596157161579615836159761, 216170961, 2761, 5761, 9361, 9961721617236173361741617476175361759617776178761783618016178961811618236183161847618, 161871618736187761879618896190161907619136193161933619496195161973619796198761993619976199962003620116201762027620296203962053620, 962081620836208762089620996211162113621296213162137621416214362153621, 1621796220362207622136222162237622396224362251622, 962273622816228762293,...

P’(7) = 2, 3, 5, 11, 13, 19, 29, 23, 31, 41, 43, 53, 59, 61, 71, 101, 79, 83, 89, 103, 109, 113, 131, 139, 149, 151, 179, 163, 181, 191, 193, 199, 211, 223, 229, 233, 239, 241, 251, 263, 269, 271, 281, 283, 293, 311, 331, 313, 349, 353, 359, 379, 383, 389, 401, 409, 419, 421, 431, 433, 439, 443, 449, 461, 463, 479, 491, 499, 503, 509, 521, 523, 541, 563, 569, 571, 593, 599, 601, 619, 613, 631, 641, 643, 653, 659, 661, 683, 691, 733, 809, 739, 821, 811, 751, 769, 823, 829, 839, 853, 859, 863, 881, 883, 911, 919, 929, 941, 953, 971, 983, 991, 1009, 1013, 1019, 1021, 1031, 1033, 1039, 1049, 1051, 1061, 1063, 1069, 1091, 1093, 1103, 1109, 1123, 1129, 1151, 1153, 1163, 1171, 1181, 1201, 1193, 1213, 1223, 1229, 1231, 1249, 1259, 1279, 1283, 1289, 1291, 1301, 1303, 1321, 1319, 1361, 1381, 1399, 1409, 1423, 1429, 1433, 1439, 1451, 1453, 1459, 1471, 1483, 1481, 1489, 1493, 1499, 1511, 1523, 1531, 1543, 1549, 1553, 1559, 1571, 1579, 1583, 1601, 1613, 1609, 1619, 1663, 1621, 1669, 1693, 1699, 1721, 1733, 1801, 1811, 1823, 1831, 1861, 1871, 1879, 1889, 1901, 1913, 1931, 1933, 1949, 1951, 1979, 1993, 1999, 2003, 2011, 2029, 2039, 2053, 2063, 2069, 2081, 2083, 2089, 2099, 2111, 2113, 2129, 2131, 2141, 2143, 2153, 2161, 2179, 2203, 2221, 2239, 2213, 2243, 2269, 2251, 2281, 2293, 2309, 2311, 2333, 2339, 2341, 2351, 2371, 2381, 2383, 2389, 2393, 2399, 2411, 2423, 2441, 2459, 2503, 2521, 2531, 2539, 2543, 2549, 2551, 2579, 2591, 2593, 2609, 2621, 2659, 2633, 2663, 2671, 2683, 2689, 2699, 2693, 2749, 2791, 2801, 2803, 2819, 2833, 2843, 2851, 2879, 2861, 2903, 2909, 2939, 2953, 2963, 2969, 2971, 3001, 2999, 3011, 3019, 3023, 3041, 3049, 3061, 3079, 3083, 3089, 3109}

Q’(7) = 273757117137197297237317417437537597617, 171017, 9783789710371097113713171397149715171, 97163718171917193719972117223722972337239724172517263726972, 172817283729373117331731373497353735973, 9738373897401740974197421743174337439744374497461746374, 974917499750375097521752375417563756975, 175937599760176197613763176417643765376597661768376917, 3378097, 39782178117, 517, 69782378297839785378597863788178837911791979297941795379, 179837991710097101371019710217103171033710397104971051710617106371069710917109371103711097112371129711517115371163711, 1711817120171193712137122371229712317124971259712, 9712837128971291713017130371321713197136171381713997140971423714297143371439714517145371459714, 1714837148171489714937149971511715237153171543715497155371559715, 1715, 97158371601716137160971619716637162171669716937169971, 2171, 337180171811718237183171861718, 1718, 971889719017191371931719337194971951719, 9719937199972003720117202972039720537206372069720817208372089720997211172113721297213172141721437215372161721, 9722037222172239722137224372269722517228172293723097231172333723397234172351723, 172381723837238972393723997241172423724417245972503725217253172539725437254972551725, 972591725937260972621726597263372663726, 17268372689726997269372, 4972, 91728017280372819728337284372851728, 972861729037290972939729537296372969729, 17300172999730117301973023730417304973061730, 9730837308973109,...

P’(8) = 2, 5, 3, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 89, 97, 101, 103, 107, 109, 113, 127, 131, 137, 139, 149, 151, 157, 163, 167, 173, 179, 181, 193, 197, 191, 199, 211, 223, 227, 229, 233, 239, 241, 251, 257, 263, 269, 271, 277, 281, 293, 307, 311, 313, 317, 331, 337, 347, 349, 353, 359, 367, 373, 379, 389, 397, 401, 409, 419, 421, 431, 433, 439, 443, 449, 457, 461, 463, 467, 479, 491, 499, 503, 509, 521, 523, 541, 547, 557, 563, 569, 571, 577, 593, 599, 601, 607, 613, 617, 619, 631, 641, 643, 647, 653, 659, 661, 673, 677, 691, 701, 709, 719, 727, 733, 739, 743, 751, 757, 761, 769, 773, 797, 821, 827, 839, 857, 863, 877, 911, 907, 919, 929, 937, 941, 947, 953, 967, 971, 977, 991, 997, 1009, 1013, 1019, 1021, 1031, 1033, 1039, 1049, 1051, 1061, 1063, 1069, 1091, 1093, 1097, 1103, 1109, 1117, 1123, 1129, 1151, 1153, 1163, 1171, 1181, 1217, 1193, 1201, 1213, 1223, 1229, 1231, 1237, 1249, 1259, 1277, 1279, 1289, 1291, 1297, 1301, 1303, 1307, 1319, 1321, 1327, 1361, 1367, 1373, 1381, 1423, 1399, 1409, 1427, 1429, 1439, 1433, 1451, 1447, 1453, 1459, 1471, 1481, 1489, 1499, 1511, 1493, 1523, 1531, 1543, 1549, 1553, 1559, 1567, 1571, 1579, 1597, 1601, 1607, 1609, 1613, 1621, 1619, 1627, 1637, 1657, 1663, 1667, 1669, 1693, 1697, 1699, 1709, 1721, 1723, 1733, 1741, 1747, 1753, 1759, 1777, 1789, 1877, 1901, 1907, 1913, 1931, 1933, 1949, 1951, 1973, 1979, 1993, 1997, 1999, 2003, 2011, 2017, 2027, 2029, 2039, 2053, 2063, 2069, 2081, 2089, 2111, 2099, 2113, 2129, 2131, 2137, 2141, 2143, 2153, 2161, 2203, 2179, 2207, 2213, 2221, 2237, 2239, 2243, 2251, 2267, 2269, 2273, 2281}

Q’(8) = 28583878118138178198238298318378418438478538598618678718738798, 9897810181038107810981138127813181378139814981518157816381678173817981, 18193819781918199821182238227822982338239824182518257826382698271827782, 18293830783118313831783318337834783498353835983678373837983, 983978401840984198421843184338439844384498457846184638467847984918499850385098521852385418547855785638569857185778593859986018607861386178619863186418643864786538659866186738677869187018709871987278733873987438751875787618769877387978, 218, 278, 398, 578, 638, 778911890789198929893789418947895389678971897789918997810098101381019810218103181033810398104981051810618106381069810918109381097811038110981117811238112981151811538116381171811, 1812178119381201812138122381229812318123781249812598127781279812, 98129181297813018130381307813198132181327813618136781373813, 1814238139981409814278142981439814338145181447814538145981471814, 1814, 981499815118149381523815318154381549815538155981567815718157981597816018160781609816138162181619816278163781657816638166781669816938169781699817098172181723817338174181747817538175981777817, 981, 77819018190781913819318193381949819518197381979819938199781999820038201182017820278202982039820538206382069820, 1820, 982111820998211382129821318213782141821438215382161822038217982207822138222182237822398224382251822678226982273822, 1,...

P’(9) = 2, 5, 3, 7, 11, 13, 17, 23, 31, 37, 41, 43, 47, 53, 61, 67, 71, 73, 101, 83, 103, 107, 113, 127, 131, 137, 151, 157, 163, 167, 173, 181, 191, 223, 211, 227, 233, 241, 251, 257, 263, 271, 277, 281, 283, 307, 311, 313, 317, 331, 337, 347, 353, 367, 373, 383, 401, 421, 431, 433, 443, 457, 461, 463, 467, 487, 491, 503, 521, 523, 541, 557, 547, 563, 571, 577, 587, 601, 607, 613, 617, 631, 641, 643, 647, 653, 661, 673, 677, 683, 691, 701, 727, 733, 743, 757, 751, 761, 773, 787, 811, 823, 821, 827, 853, 857, 863, 877, 881, 883, 887, 977, 1013, 1031, 1021, 1033, 1051, 1061, 1063, 1091, 1087, 1117, 1103, 1123, 1151, 1153, 1163, 1171, 1181, 1187, 1201, 1213, 1217, 1223, 1231, 1237, 1277, 1283, 1291, 1301, 1303, 1307, 1321, 1327, 1361, 1367, 1373, 1381, 1423, 1427, 1433, 1447, 1451, 1453, 1471, 1481, 1483, 1487, 1511, 1523, 1531, 1543, 1553, 1567, 1571, 1583, 1601, 1607, 1613, 1621, 1627, 1637, 1657, 1663, 1667, 1721, 1723, 1733, 1741, 1747, 1753, 1777, 1783, 1787, 1801, 1811, 1823, 1831, 1847, 1861, 1867, 1871, 1873, 1877, 1933, 2003, 1951, 1987, 2011, 2017, 2027, 2053, 2063, 2081, 2083, 2087, 2111, 2113, 2131, 2141, 2137, 2143, 2153, 2161, 2203, 2207, 2213, 2221, 2237, 2243, 2251, 2267, 2273, 2281, 2311, 2287, 2333, 2341, 2347, 2351, 2357, 2371, 2377, 2381, 2383, 2411, 2417, 2423, 2437, 2441, 2447, 2467, 2473, 2477, 2503, 2521, 2531, 2543, 2551, 2557, 2591, 2621, 2617, 2633, 2647, 2657, 2663, 2671, 2677, 2683, 2687, 2707, 2711, 2713, 2731, 2741, 2753, 2767, 2777, 2791, 2801, 2803, 2833, 2837, 2843, 2851, 2857, 2861, 2887, 2927, 2957, 2963, 3001, 3011, 3023, 3037, 3041, 3061, 3067, 3083, 3121, 3137, 3163, 3167, 3181, 3187, 3191, 3203, 3217, 3221}

Q’(9) = 2959397911913917923931937941943947953961967971973910198391039107911391279131913791519157916391679173918191, 192239211922792339241925192579263927192779281928393079311931393179331933793479353936793739383940194219431943394439457946194639467948794, 19503952195239541955795479563957195779587960196079613961796319641964396479653966196739677968396, 1970197279733974397579751976197739787981198239821982798539857986398779881988398879, 7791013910319102191033910519106191063910, 1910879111791103911239115191153911639117191181911879120191213912179122391231912379127791283912, 19130191303913079132191327913619136791373913819142391427914339144791451914539147191481914839148791511915239153191543915539156791571915839160191607916139162191627916379165791663916679172191723917339174191747917539177791783917879180191811918239183191847918619186791871918739187791, 339200391, 5191, 8792011920179202792053920639208192083920879211192113921319214192137921439215392161922039220792213922219223792243922519226792273922819231192287923339234192347923519235792371923779238192383924119241792423924379244192447924679247392477925039252192531925439255192557925, 1926219261792633926479265792663926719267792683926879270792711927139273192741927539276792777927, 192801928039283392837928439285192857928619288792, 2792, 5792, 639300193011930239303793041930619306793083931219313793163931679318193187931, 1932039321793221,...

P’(0) = 2, 3, 5, 7, 11, 13, 17, 19, 29, 23, 31, 37, 41, 47, 43, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 101, 113, 109, 131, 127, 137, 139, 149, 151, 163, 157, 167, 173, 179, 181, 191, 193, 197, 199, 211, 223, 227, 229, 233, 239, 241, 251, 257, 263, 269, 271, 277, 281, 283, 293, 311, 313, 317, 331, 337, 347, 349, 353, 359, 367, 373, 379, 383, 389, 397, 401, 409, 419, 421, 433, 431, 439, 443, 449, 457, 461, 463, 467, 479, 487, 491, 499, 509, 521, 523, 547, 541, 557, 563, 569, 571, 577, 587, 593, 599, 601, 617, 613, 619, 631, 641, 643, 647, 653, 659, 661, 673, 677, 683, 691, 701, 719, 727, 709, 733, 739, 743, 751, 757, 761, 769, 773, 787, 797, 809, 811, 821, 823, 827, 829, 839, 853, 857, 859, 863, 877, 881, 883, 887, 911, 919, 929, 937, 941, 947, 953, 967, 971, 977, 983, 991, 997, 1021, 1033, 1109, 1039, 1049, 1051, 1117, 1063, 1123, 1069, 1087, 1091, 1093, 1129, 1151, 1153, 1163, 1171, 1181, 1187, 1193, 1201, 1213, 1223, 1217, 1229, 1231, 1237, 1249, 1259, 1277, 1279, 1283, 1289, 1291, 1297, 1301, 1319, 1321, 1327, 1361, 1367, 1373, 1381, 1399, 1409, 1433, 1423, 1427, 1429, 1439, 1447, 1451, 1453, 1459, 1471, 1481, 1483, 1487, 1489, 1493, 1499, 1511, 1523, 1531, 1543, 1549, 1553, 1559, 1567, 1571, 1579, 1583, 1597, 1601, 1609, 1619, 1613, 1621, 1627, 1637, 1657, 1663, 1667, 1669, 1693, 1697, 1699, 1709, 1721, 1723, 1733, 1741, 1747, 1753, 1759, 1783, 1777, 1787, 1789, 1801, 1811, 1823, 1831, 1847, 1861, 1867, 1871, 1873, 1877, 1879, 1889, 1901, 1913, 1933, 1931, 1949, 1951, 1973, 1979, 1987, 1993, 1997, 1999, 2027, 2039, 2063, 2069, 2081, 2087, 2099, 2111, 2113, 2131, 2129, 2137, 2141, 2153,...

Q’(0) = 203050701101301701902902303103704104704305305906106707107307908308909701, 1011301, 90131012701370139014901510163015701670173017901810191019301970199021102230227022902330239024102510257026302690271027702810283029303110313031703310337034703490353035903670373037903830389039704, 104, 904190421043304310439044304490457046104630467047904870491049905, 905210523054705410557056305690571057705870593059906, 10617061306190631064106430647065306590661067306770683069107, 10719072707, 9073307390743075107570761076907730787079708, 908110821082308270829083908530857085908630877088108830887091109190929093709410947095309670971097709830991099701, 2101, 33011, 901, 3901, 4901, 510111701, 630112301, 6901, 8701, 9101, 930112901151011530116301171011810118701193012, 10121301223012170122901231012370124901259012770127901283012890129101297013, 10131901321013270136101367013730138101399014, 901433014230142701429014390144701451014530145901471014810148301487014890149301499015110152301531015430154901553015590156701571015790158301597016, 1016, 9016190161301621016270163701657016630166701669016930169701699017, 90172101723017330174101747017530175901783017770178701789018, 10181101823018310184701861018670187101873018770187901889019, 1019130193301931019490195101973019790198701993019970199902, 2702, 3902, 6302, 6902, 8102, 8702, 9902111021130213102129021370214102153}

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Merci à tous !

Thank you Lars, Aai and Jean-Marc!