Two sum up to a prime? Erase!

 

Hello SeqFans,

 

Submit this sequence to a teenager and ask him to underline all couples of neighboring integers which sum up to a prime:

 

S = 1,2,4,3,1,3,2,2,6,5,4,2,11,3,3,14,1,7,12,3,1,22,2,4,25,2,2,29,6,...

 

He will underline 1,2, then 4,3, then 3,2, then 6,5, etc. Ė like this:

 

S : 1,2,4,3,1,3,2,2,6,5,4,2,11,3,3,14,1,7,12,3,1,22,2,4,25,2,2,29,6,...

††† *** ***†† ***†† ***†† ****†† ****†† ****†† ****†† ****†† ****

††††††††††† 1†††† 2†††† 4††††† 3††††† 1††††† 3††††† 2††††† 2††††† 6...

 

Now for the first amazing remark:

- all "prime sums" are different and they seem to be a derangement

of the prime numbers (they are);

And the second remark:

- the not underlined integers re-build the sequence S itself.

 

We have thus a fractal sequence.

 

(...)

 

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After a few mails with Franklin T. Adams-Watters, Iíve decided to drop the ďfractalĒ claim and to admit zeros in the sequence.

 

Lars Blomberg then came with this perfect new S:

 

> ę Hello Eric,

††Iíve modified the program to generate the new sequence

 

Start with:

1,1

Sequence:

S = 1, 1, 0, 3, 1, 0, 5, 1, 3, 4, 0, 0, 11, 3, 5, 8, 1, 3, 14, 0, 0, 19, 5, 3, 20, 1, 0, 29, 3, 1, 30, 4, 2, 35, 0, 1, 40, 0, 0, 43, 11, 1, 46, 3, 1, 52, 5, 1, 58, 8, 0, 61, 1, 0, 67, 3, 1, 70, 14, 1, 72, 0, 1, 78, 0, 0, 83, 19, 1, 88, 5, 1, 96, 3, 1, 100, 20, 0, 103, 1, 3, 104, 0, 0, 109, 29, 1, 112, 3, 3, 124, 1, 0, 131, 30, 0, 137, 4, 0, 139, 2, 2, 147, 35, 1, 150, 0, 0, 157, 1, 0, 163, 40, 2, 165, 0, 1, 172, 0, 0, 179, 43, 2, 179, 11, 3, 188, 1, 3, 190, 46, 0, 197, 3, 5, 194, 1, 3, 208, 52, 0, 223, 5, 3, 224, 1, 0, 229, 58, 4, 229, 8, 1, 238, 0, 0, 241, 61, 2, 249, 1, 3, 254, 0, 0, 263, 67, 2, 267, 3, 5, 266, 1, 3, 274, 70, 0, 281, 14, 0, 283, 1, 0, 293, 72, 2, 305, 0, 0, 311, 1, 0, 313, 78, 2, 315, 0, 1, 330, 0, 0, 337, 83, 1, 346, 19, 2, 347, 1, 0, 353, 88, 0, 359, 5, 3, 364, 1, 0, 373, 96, 0, 379, 3, 3, 380, 1, 0, 389, 100, 0, 397, 20, 1, 400, 0, 0, 409, 103, 2, 417, 1, 0, 421, 3, 1, 430, 104, 1, 432, 0, 1, 438, 0, 0, 443, 109, 1, 448, 29, 3, 454, 1, 0, 461, 112, 0, 463, 3, 1, 466, 3, 1, 478, 124, 0, 487, 1, 3, 488, 0, 0, 499, 131, 1, 502, 30, 2, 507, 0, 0, 521, 137, 1, 522, 4, 2, 539, 0, 0, 547, 139, 1, 556, 2, 4, 559, 2, 2, 567, 147, 0, 571, 35, 0, 577, 1, 0, 587, 150, 2, 591, 0, 1, 598, 0, 0, 601, 157, 2, 605, 1, 3, 610, 0, 0, 617, 163, 1, 618, 40, 0, 631, 2, 2, 639, 165, 1, 642, 0, 0, 647, 1, 0, 653, 172, 2, 657, 0, 1, 660, 0, 0, 673, 179, 3, 674, 43, 1, 682, 2, 2, 689, 179, 1, 700, 11, 1, 708, 3, 1, 718, 188, 0, 727, 1, 0, 733, 3, 1, 738, 190, 0, 743, 46, 2, 749, 0, 0, 757, 197, 1, 760, 3, 3, 766, 5, 1, 772, 194, 0, 787, 1, 0, 797, 3, 1, 808, 208, 1, 810, 52, 2, 819, 0, 0, 823, 223, 1, 826, 5, 1, 828, 3, 1, 838, 224, 0, 853, 1, 3, 854, 0, 0, 859, 229, 1, 862, 58, 2, 875, 4, 0, 881, 229, 1, 882, 8, 0, 887, 1, 0, 907, 238, 2, 909, 0, 1, 918, 0, 0, 929, 241, 2, 935, 61, 1, 940, 2, 2, 945, 249, 0, 953, 1, 0, 967, 3, 1, 970, 254, 1, 976, 0, 1, 982, 0, 0, 991, 263, 2, 995, 67, 1, 1008, 2, 2, 1011, 267, 0, 1019, 3, 1, 1020, 5, 1, 1030, 266, 0, 1033, 1, 0, 1039, 3, 1, 1048, 274, 0, 1051, 70, 2, 1059, 0, 0, 1063, 281, 1, 1068, 14, 1, 1086, 0, 0, 1091, 283, 2, 1091, 1, 3, 1094, 0, 0, 1103, 293, 1, 1108, 72, 0, 1117, 2, 2, 1121, 305, 1, 1128, 0, 1, 1150, 0, 0, 1153, 311, 3, 1160, 1, 3, 1168, 0, 0, 1181, 313, 1, 1186, 78, 0, 1193, 2, 2, 1199, 315, 1, 1212, 0, 0, 1217, 1, 3, 1220, 330, 2, 1227, 0, 1, 1230, 0, 0, 1237, 337, 1, 1248, 83, 2, 1257, 1, 0, 1277, 346, 0, 1279, 19, 1, 1282, 2, 2, 1287, 347, 4, 1287, 1, 3, 1294, 0, 0, 1301, 353, 1, 1302, 88, 2, 1305, 0, 0, 1319, 359, 1, 1320, 5, 1, 1326, 3, 1, 1360, 364, 0, 1367, 1, 3, 1370, 0, 0, 1381, 373, 1, 1398, 96, 2, 1407, 0, 0, 1423, 379, 2, 1425, 3, 1, 1428, 3, 1, 1432, 380, 0, 1439, 1, 3, 1444, 0, 0, 1451, 389, 1, 1452, 100, 2, 1457, 0, 0, 1471, 397, 1, 1480, 20, 0, 1483, 1, 0, 1487, 400, 3, 1486, 0, 1, 1492, 0, 0, 1499, 409, 2, 1509, 103, 1, 1522, 2, 2, 1529, 417, 0, 1543, 1, 3, 1546, 0, 0, 1553, 421, 1, 1558, 3, 3, 1564, 1, 0, 1571, 430, 0, 1579, 104, 0, 1583, 1, 3, 1594, 432, 2, 1599, 0, 0, 1607, 1, 0, 1609, 438, 2, 1611, 0, 1, 1618, 0, 0, 1621, 443, 1, 1626, 109, 2, 1635, 1, 0, 1657, 448, 0, 1663, 29, 4, 1663, 3, 1, 1668, 454, 0, 1693, 1, 3, 1694, 0, 0, 1699, 461, 1, 1708, 112, 2, 1719, 0, 0, 1723, 463, 1, 1732, 3, 3, 1738, 1, 3, 1744, 466, 0, 1753, 3, 3, 1756, 1, 0, 1777, 478, 2, 1781, 124, 1, 1786, 0, 0, 1789, 487, 2, 1799, 1, 0, 1811, 3, 1, 1822, 488, 1, 1830, 0, 1, 1846, 0, 0, 1861, 499, 2, 1865, 131, 2, 1869, 1, 0, 1873, 502, 2, 1875, 30, 0, 1879, 2, 2, 1887, 507, 1, 1900, 0, 1, 1906, 0, 0, 1913, 521, 1, 1930, 137, 4, 1929, 1, 0, 1949, 522, 0, 1951, 4, 0, 1973, 2, 2, 1977, 539, 1, 1986, 0, 1, 1992, 0, 0, 1997, 547, 2, 1997, 139, 2, 2001, 1, 0, 2011, 556, 0, 2017, 2, 6, 2021, 4, 0, 2029, 559, 0, 2039, 2, 4, 2049, 2, 2, 2061, 567, 0, 2069, 147, 1, 2080, 0, 0, 2083, 571, 1, 2086, 35, 1, 2088, 0, 0, 2099, 577, 2, 2109, 1, 3, 2110, 0, 0, 2129, 587, 1, 2130, 150, 0, 2137, 2, 2, 2139, 591, 1, 2142, 0, 0, 2153, 1, 0, 2161, 598, 2, 2177, 0, 1, 2202, 0, 0, 2207, 601, 1, 2212, 157, 1, 2220, 2, 2, 2235, 605, 0, 2239, 1, 0, 2243, 3

Primes used:

2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 101, 103, 107, 109, 113, 127, 131, 137, 139, 149, 151, 157, 163, 167, 173, 179, 181, 191, 193, 197, 199, 211, 223, 227, 229, 233, 239, 241, 251, 257, 263, 269, 271, 277, 281, 283, 293, 307, 311, 313, 317, 331, 337, 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, 523, 541, 547, 557, 563, 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, 727, 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, 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, 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, 1409, 1423, 1427, 1429, 1433, 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, 1607, 1609, 1613, 1619, 1621, 1627, 1637, 1657, 1663, 1667, 1669, 1693, 1697, 1699, 1709, 1721, 1723, 1733, 1741, 1747, 1753, 1759, 1777, 1783, 1787, 1789, 1801, 1811, 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, 2053, 2063, 2069, 2081, 2083, 2087, 2089, 2099, 2111, 2113, 2129, 2131, 2137, 2141, 2143, 2153, 2161, 2179, 2203, 2207, 2213, 2221, 2237, 2239, 2243, 2251

 

... and Iíve verified that the primes now come in order. This version appeals to me: because of this and because it starts with just two 1ís. Also, the modified rules are simpler. Ľ

 

Thank you, Lars Ė good job!

 

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From: Eric Angelini

Sent: Thursday, September 22, 2011 1:32 PM

To: franktaw@netscape.net

Cc: Lars Blomberg

Subject: "Prime sums" of underlined neighbors -- Ultimate try (?)

 

Hello again, Franklin,

(...)

If we admit that a(n) can be equal to zero, than S could show _from the beginning_ the succession of the "prime sums" as in A000040:

 

 

1,1,0,3,1,0,5,1,3,4,0,0,11,3,5,8,1,3,14,0,0,19,5,3,20,1,0,29,3,1,30,4,2,35,0,1,40,

S=1,1,0,3,1,0,5,1,3,4,0,0,11,3,5,8,1,3,14,0,0,19,5,3,20,1,0,29,3,1,30,4,2,35,0,1,40,0,0,43,11,...

†† 2†† 3†††† 5†††† 7†††† 11†††† 13††† 17†††† 19†††† 23†††† 29†††† 31†††† 37†††† 41†††† 43

††††††††† 1†††† 1†††† 0††††† 3†††† 1††††† 0††††† 5††††† 1††††† 3††††† 4††††† 0††††† 0††††† 11...

 

This seems to be the lexico-first such sequence where zeroes are allowed as the start 0,2,2,1 doesnít work:

 

T =0,2,2,1,0,1,4,2,?,?,2,

††††† 2†† 3†††† 5†††† 7

†††††††††††† 0†††† 2†††† 2

 

Iím fairly confident that all "prime-sums" will show as it is easier and easier to find two integers summing up to a bigger and bigger prime.

Best,

….