Brussels, (Belgium), Feb. 22^nd 2006

Hello everyone,

This is the copy of an e-mail I’ve sent to Neil a week or so ago -- and Neil’s answer.

Is there something interesting in this?

Best,

É.

------------

[Eric A.] :

> Hello Neil,

I've had an idea about prime numbers which I would like to submit to you (if you have a minute):

We could see the prime numbers sequence as the result of an algorithm like this:

a) A(1)=2 (this is the seed);

b) next integer must be strictly superior than the previously written one;

c) this integer must be the first available one which is not a multiple of one of the previous integers in the sequence.

With those rules we produce A000040 [or S(1)]:

S(1) = 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...

My idea, yesterday, was to change slightly the (c) rule, and put "second" instead of "first"; this gives the c' rule:

c') this integer must be the *second* available one which is not a multiple of one of the previous integers in the sequence.

That would lead to S(2) - which is not in the OEIS (if I didn't make mistakes in computing the terms by hand):

S(2) = 2,5,9,13,19,23,31,37,43,49,53,61,71,77,83,89,101...

Explanation:

2 is the seed; we have:

S(2) = 2, ...

- what comes next? Can we write "3" after 2? No because 3 is the *first* available integer which is not a multiple of 2 -- we don't want it, we are looking for the *second* such integer;

- is it "4"? No, 4 is even (thus a multiple of 2);

- is it "5"? Yes, 5 is not a multiple of 2 -- and "5" is indeed our *second* choice; we thus have now:

S(2) = 2, 5, ...

- what comes next? Can we write "6" after 5? No, because 6 is even;

- can we put "7"? No, because 7 is the *first* available integer -- and we don't want it, we are looking for the *second* such integer;

- is it "8"? No, 8 is even;

- is it "9"? Yes, 9 is not a multiple of 2 nor 5 -- and "9" is indeed our *second* choice; we thus have now:

S(2) = 2, 5, 9, ...

- what comes next? Can we write "10" after 9? No, because 10 is even;

- can we put "11"? No, because 11 is the *first* available integer -- and we don't want it, we are looking for the *second* such integer;

- is it "12"? No, 12 is even;

- is it "13"? Yes, 13 is not a multiple of 2, nor 5, nor 9, and "13" is indeed our *second* choice; we thus have now:

S(2) = 2, 5, 9, 13, ...

- could we put "14" after 13 ? No, 14 is even;

- could we put "15" ? No, 15 is a multiple of 5;

- could we put "16" ? No, 16 is even;

- could we put "17" ? No, because 17 is the *first* available integer -- and we don't want it, we are looking for the *second* such integer;

- is it "18"? No, 18 is even;

- is it "19"? Yes, 19 is not a multiple of 2, nor 5, nor 9, nor 13, and "19" is indeed our *second* choice; we thus have now:

S(2) = 2, 5, 9, 13, 19, ...

etc.

Now the big thing (for me -- this might be old hat for you!): the generalization.

We could have produced the c'' rule in the same way, saying:

c'') this integer must be the *third* available one which is not a multiple of one of the previous integers in the sequence.

This would lead to S(3) - which is not in the OEIS either:

S(3) = 2,7,13,19,27,33,43,51, ...

etc.

So the general "c-k" rule would say:

c-k) this integer must be the *k-th* available one which is not a multiple of one of the previous integers in the sequence.

We see now that the primes sequence A000040 is only a particular case of the general rule, with k=1!

Isn't this something? ;-)

Best,

É.

-----------------------------------------------------------

[Neil] :

> That's a nice idea! Can you send in the sequences for k=2, 3 and 4, say?

It will be interesting to study these sequences.

Neil

-----------------------------------------------------------

Ok, here it is for k=1 (S1) to k=10 (S10) [terms computed by Gilles Sadowski]:

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=S1 (primes ; A000040)

2,5,9,13,19,23,31,37,43,49,53,61,71,77,83,89,101,107,113,121,127,137,141,151,163,173,179,187,193,199,203,219,227,233,239,251,263,271,281,287,291,307,311,317,327,331,347,353,359,373,383,393,401,413,419,431,439,447,451,461,467,471,487,493,501,509,517,523,543,553,563,571,577,591,599,607,617,631,641,647,653,661,673,679,687,697,709,721,727,737,743,757,763,771,787,799,807,811,823,829=S2

2,7,13,19,27,33,43,51,59,69,75,85,93,103,111,121,127,139,149,157,167,181,187,197,205,219,229,237,249,257,265,271,281,291,305,313,321,335,341,353,365,373,389,395,407,415,423,435,445,453,463,479,489,499,509,521,529,541,549,565,573,579,593,601,613,625,633,643,655,667,673,683,697,705,717,725,737,751,757,773,787,799,807,815,827,839,851,859,869,881,895,907,913,925,937,947,955,967,979,991=S3

2,9,17,25,35,43,55,65,73,87,95,107,115,131,141,149,159,169,181,193,203,213,229,237,249,259,269,281,293,305,313,329,339,349,361,373,383,399,409,417,429,439,451,463,471,485,497,505,519,533,545,555,569,581,591,601,615,627,637,649,661,673,683,695,707,719,733,743,757,767,779,789,809,821,831,847,857,865,881,889,899,919,931,943,955,971,983,995,1007,1019,1029,1041,1057,1067,1079,1093,1103,1117,1129,1141=S4

2,11,21,31,43,53,67,79,89,103,115,127,139,153,169,179,193,205,221,233,245,257,271,287,299,313,327,339,353,367,379,397,413,425,437,453,467,485,497,509,523,541,551,569,585,597,613,629,641,657,669,681,697,709,727,743,753,767,785,797,815,831,843,857,873,883,907,919,931,947,959,975,993,1009,1021,1041,1053,1065,1081,1095,1109,1123,1135,1153,1171,1187,1203,1215,1231,1245,1261,1279,1295,1305,1319,1341,1361,1373,1387,1403=S5

2,13,25,37,51,63,79,93,105,121,135,149,163,179,197,209,223,239,253,269,287,301,313,329,343,359,373,389,409,421,437,453,471,487,501,515,529,545,565,581,597,613,631,645,659,677,691,707,727,743,757,779,791,805,823,839,857,879,891,909,921,937,955,967,983,1003,1015,1035,1051,1065,1087,1101,1117,1135,1151,1169,1189,1205,1223,1241,1255,1273,1289,1303,1319,1337,1357,1381,1397,1409,1427,1441,1457,1481,1497,1515,1533,1549,1563,1583=S6

2,15,29,43,59,73,91,107,121,139,155,171,187,205,223,239,253,271,289,307,327,341,357,379,395,411,429,445,461,479,497,515,533,549,571,587,601,621,635,655,677,691,709,727,745,763,785,801,823,839,859,875,891,909,927,943,965,985,1007,1027,1043,1057,1079,1099,1113,1135,1151,1169,1191,1209,1225,1249,1267,1283,1299,1317,1337,1351,1373,1393,1409,1429,1449,1465,1483,1501,1519,1543,1563,1581,1603,1619,1633,1651,1669,1691,1709,1731,1745,1765=S7

2,17,33,49,67,83,103,121,137,157,175,193,211,229,251,269,285,305,325,345,365,381,399,419,439,457,479,499,517,537,555,573,593,615,639,655,673,695,713,733,757,777,795,815,835,853,877,895,917,939,955,981,997,1017,1041,1059,1077,1101,1119,1141,1163,1181,1199,1217,1239,1263,1283,1301,1321,1347,1367,1387,1403,1431,1451,1467,1489,1509,1531,1555,1571,1595,1619,1637,1655,1673,1693,1709,1735,1761,1777,1799,1823,1841,1861,1877,1901,1929,1949,1971=S8

2,19,37,55,75,93,115,135,153,175,195,215,235,255,277,299,317,339,359,381,403,425,445,467,489,509,531,549,571,595,615,637,659,681,699,723,743,761,785,803,827,849,871,891,913,937,959,981,1001,1025,1047,1065,1093,1111,1133,1157,1179,1199,1223,1245,1263,1287,1307,1327,1347,1371,1399,1419,1439,1459,1481,1507,1529,1549,1567,1599,1619,1637,1661,1681,1707,1731,1751,1773,1793,1819,1837,1859,1879,1901,1927,1949,1975,1999,2019,2045,2063,2083,2105,2129=S9

2,21,41,61,83,103,127,149,169,193,215,237,259,281,303,329,349,373,395,419,443,467,489,511,537,559,585,605,629,655,677,701,725,751,771,797,817,839,865,885,911,937,961,983,1005,1027,1055,1081,1101,1127,1151,1175,1203,1223,1247,1269,1293,1315,1343,1369,1389,1421,1443,1465,1487,1511,1541,1563,1587,1609,1633,1663,1687,1709,1729,1759,1783,1805,1831,1853,1879,1903,1931,1959,1985,2007,2033,2055,2077,2105,2129,2153,2181,2203,2227,2259,2279,2301,2325,2351=S10

...

---------------

S1: 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,

S2: 2, 5, 9,13,19, 23, 31, 37, 43, 49, 53, 61, 71, 77, 83, 89,101,107,113,121,127,137,141,151,163,173,179,187,193,199,203,219,227,233,239,251,

S3: 2, 7,13,19,27, 33, 43, 51, 59, 69, 75, 85, 93,103,111,121,127,139,149,157,167,181,187,197,205,219,229,237,249,257,265,271,281,291,305,313,

S4: 2, 9,17,25,35, 43, 55, 65, 73, 87, 95,107,115,131,141,149,159,169,181,193,203,213,229,237,249,259,269,281,293,305,313,329,339,349,361,373,

S5: 2,11,21,31,43, 53, 67, 79, 89,103,115,127,139,153,169,179,193,205,221,233,245,257,271,287,299,313,327,339,353,367,379,397,413,425,437,453,

S6: 2,13,25,37,51, 63, 79, 93,105,121,135,149,163,179,197,209,223,239,253,269,287,301,313,329,343,359,373,389,409,421,437,453,471,487,501,515,

S7: 2,15,29,43,59, 73, 91,107,121,139,155,171,187,205,223,239,253,271,289,307,327,341,357,379,395,411,429,445,461,479,497,515,533,549,571,587,

S8: 2,17,33,49,67, 83,103,121,137,157,175,193,211,229,251,269,285,305,325,345,365,381,399,419,439,457,479,499,517,537,555,573,593,615,639,655,

S9: 2,19,37,55,75, 93,115,135,153,175,195,215,235,255,277,299,317,339,359,381,403,425,445,467,489,509,531,549,571,595,615,637,659,681,699,723,

S10: 2,21,41,61,83,103,127,149,169,193,215,237,259,281,303,329,349,373,395,419,443,467,489,511,537,559,585,605,629,655,677,701,725,751,771,797,

...

---------------

At first glance vertical patterns appear... but disappear soon if you investigate further...

Best,

É.