One, two, three... thousand Zeta functions!

One can produce the prime numbers sequence with this simple algorithm (Eratosthenes):

1) Start examining “2”

2) Call “atom” the first integer which cannot be divided by another atom

3) Examine the next natural integer

4) Go to (2)

Let’s start P, having N (the natural numbers) in mind:

N=2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,...

P=2

Is “2” an atom? Yes, because “2” cannot be divided by another atom (there are no such atoms yet, thus no division!)

So we “yellow” this integer in P (yellow = atom = prime number) and we immediately forbid the use of any multiple of 2 (marked here with an “F” for “Forbidden”):

N=2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,...

P=2,3,F,5,F,7 F,9, F,11, F,13, F,15, F,17, F,19, F,21, F,23, F,25, F,27, F,29, F,31, F,33, F,35, F,37, F,39, F,41, F,43, F,45, F,47, F,

Next integer to be examined is “3”; is “3” an atom? Yes, because 3 is not divisible by another atom (3 is not divisible by 2). P will look like this now (after “yellowing” 3 and forbidding all multiples of 3):

N=2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,...

P=2,3,F,5,F,7 F,F, F,11, F,13, F, F, F,17, F,19, F, F, F,23, F,25, F, F, F,29, F,31, F, F, F,35, F,37, F, F, F,41, F,43, F, F, F,47, F,

Etc. This sieving method gives all prime numbers in order:

N=2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,...

P=2,3,F,5,F,7 F,F, F,11, F,13, F, F, F,17, F,19, F, F, F,23, F, F, F, F, F,29, F,31, F, F, F, F, F,37, F, F, F,41, F,43, F, F, F,47, F,

We would like to produce now another kind of numbers: not the primes, but the “seconds”!

We will thus slightly change the generating algorithm into:

1) Start examining “2”

2) Call “atom” the second integer which cannot be divided by another atom

3) Examine the next natural integer

4) Go to (2)

What did we change? The second step -- which asks now to call “atom” not the first candidate, but the second integer in N which cannot be divided by another atom -- thus the second candidate:

Let’s start S (noting “F” for forbidden as before, and “C” for a not selected candidate integer):

N=2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,...

S=2,

Is “2” an atom? Almost! Sure “2” cannot be divided by another atom, but “2” is the first such candidate -- and we only keep the second ones! Thus we mark a “C” under “2” in S:

N=2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,...

S=C,

Is “3” an atom? Yes, because “3” cannot be divided by another atom (there are none, yet) and “3” is now our second candidate. Thus we “yellow” it and mark accordingly with an “F” all multiples of “3”:

N=2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,...

S=C,3,4,5,F,7,8,F,10,11, F,13,14, F,16,17, F,19,20, F,22,23, F,25,26, F,28,29, F,31,32, F,34,35, F,37,38, F,40,41, F,43,44, F,46,47, F,...

Is “4” an atom? Almost, but no! (“4” cannot be divided by 3, but “4” is our first such candidate -- and we are only interested in second candidates). Thus we mark “C” under “4”, leaving untouched the rest of S:

N=2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,...

S=C,3,C,5,F,7,8,F,10,11, F,13,14, F,16,17, F,19,20, F,22,23, F,25,26, F,28,29, F,31,32, F,34,35, F,37,38, F,40,41, F,43,44, F,46,47, F,...

Is “5” an atom? Yes! (“5” cannot be divided by any other atom --here “3”--, and “5” is the next such candidate after “4”). We yellow it and mark accordingly with F all multiples of 5:

N=2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,...

S=C,3,C,5,F,7,8,F, F,11, F,13,14, F,16,17, F,19, F, F,22,23, F, F,26, F,28,29, F,31,32, F,34, F, F,37,38, F, F,41, F,43,44, F,46,47, F,...

We don’t examine “6” (forbidden, thus not candidate) and proceed to “7”; but “7” is only the first integer which cannot be divided by another atom -- and, again, we only keep the second such candidates. Thus we mark “C” under “7” -- and proceed:

N=2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,...

S=C,3,C,5,F,C,8,F, F,11, F,13,14, F,16,17, F,19, F, F,22,23, F, F,26, F,28,29, F,31,32, F,34, F, F,37,38, F, F,41, F,43,44, F,46,47, F,...

Is “8” an atom? Yes, “8” cannot be divided by 3 or by 5 (atoms) and “8” is the second available integer with this property -- we yellow it and mark with “F” all 8-multiples:

N=2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,...

S=C,3,C,5,F,C,8,F, F,11, F,13,14, F, F,17, F,19, F, F,22,23, F, F,26, F,28,29, F,31, F, F,34, F, F,37,38, F, F,41, F,43,44, F,46,47, F,...

“11” is our next candidate -- but “13” our next atom -- and the multiples of 13 the next forbidden integers marked with F:

N=2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,...

S=C,3,C,5,F,C,8,F, F, C, F,13,14, F, F,17, F,19, F, F,22,23, F, F, F, F,28,29, F,31, F, F,34, F, F,37,38, F, F,41, F,43,44, F,46,47, F,...

Going on with the algorithm, we get this sequence:

N=2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,...

S=C,3,C,5,F,C,8,F, F, C, F,13, C, F, F,17, F, C, F, F,22, C, F, F, F, F,28, C, F,31, F, F, F, F, F, C,38, F, F, C, F,43, F, F, C,47, F,...

(by definition there are exactly one C between two atoms, and a varying quantity of F’s):

S=3,5,8,13,17,22,28,31,38,43,47,... which is not (yet) in the OEIS (but will be soon)

This sequence could be called “Second numbers sequence” (instead of “prime numbers sequence”).

We could now change the second step of the algorithm into this:

1) Start examining “2”

2) Call “atom” the third integer which cannot be divided by another atom

3) Examine the next natural integer

4) Go to (2)

... and build another sequence - the “third numbers sequence”:

T=4,7,11,17,23,27,31,39,45,53,...

Varying k in step (2) of the algorithm (we have seen that k=1 produces the prime numbers, k=2 the “second numbers”, k=3 the “third numbers”) will lead to the following array (many, many thanks to Mensanator who has computed 150 integers for k=1 to k=30 and to Hugo van der Sanden who was of immense help fixing a few bugs in my brain):

k=1: 2,3,5,7,11,13,17,19,23,29,31,37,41,43,47,53,59,61,67,71,73,79,83,89,97,101,103,107,109,113,127,131,137,139,149,151,157,163,167,173,179,181,191,193,197,199,211,223,227,229,233,239,241,251,257,263,269,271,277,281,283,293,307,311,313,317,331,337,347,349,353,359,367,373,379,383,389,397,401,409,419,421,431,433,439,443,449,457,461,463,467,479,487,491,499,503,509,521,523,541,547,557,563,569,571,577,587,593,599,601,607,613,617,619,631,641,643,647,653,659,661,673,677,683,691,701,709,719,727,733,739,743,751,757,761,769,773,787,797,809,811,821,823,827,829,839,853,857,859,863,

k=2: 3,5,8,13,17,22,28,31,38,43,47,53,59,67,73,77,82,89,97,101,107,113,121,127,133,139,148,151,158,163,167,179,191,197,203,209,218,227,233,241,251,257,262,269,274,281,284,293,307,313,317,322,332,343,347,353,361,367,379,386,397,401,409,419,422,431,437,443,449,457,461,467,479,491,499,509,521,526,541,547,553,557,566,571,581,593,599,607,617,622,631,643,653,661,667,673,677,691,698,703,718,721,727,739,746,757,763,769,778,781,796,809,821,827,839,842,853,857,863,869,878,883,892,911,916,919,929,941,947,956,967,974,983,994,1006,1013,1021,1033,1043,1049,1058,1063,1073,1087,1093,1099,1108,1117,1126,1129,

k=3: 4,7,11,17,23,27,31,39,45,53,59,67,74,82,87,95,103,111,122,127,131,141,146,151,163,169,178,183,193,199,211,215,223,229,237,247,251,263,271,278,290,298,307,314,325,334,342,349,358,362,369,377,383,394,401,415,421,433,445,454,463,470,479,485,498,503,514,523,537,543,551,559,565,571,582,591,601,611,617,625,634,642,653,659,673,678,685,695,703,710,719,725,734,745,750,758,771,778,787,794,807,817,822,829,838,843,857,863,877,881,887,898,909,919,925,934,941,951,963,974,982,991,998,1011,1018,1025,1033,1041,1051,1063,1070,1079,1087,1093,1097,1109,1119,1129,1138,1153,1159,1167,1175,1187,1193,1201,1214,1226,1231,1238,

k=4: 5,9,14,21,26,33,39,46,51,59,67,73,79,87,93,101,109,116,123,129,137,143,152,163,169,178,187,193,203,212,221,227,239,247,253,259,269,278,284,293,301,311,318,328,334,343,349,359,367,377,383,391,398,409,419,427,437,446,452,461,471,482,491,498,503,514,523,529,541,547,559,566,577,583,592,599,611,618,628,634,642,653,664,673,679,688,694,703,713,722,731,743,752,761,769,778,787,793,802,814,824,833,842,849,857,866,883,889,899,907,916,923,932,941,951,967,974,983,992,1004,1013,1021,1034,1041,1049,1057,1067,1079,1086,1093,1103,1117,1126,1133,1146,1156,1164,1174,1182,1191,1202,1211,1219,1229,1238,1252,1261,1267,1282,1291,

k=5: 6,11,17,25,31,39,46,56,63,70,79,86,95,104,111,122,130,137,146,152,163,171,181,191,199,206,214,226,235,245,256,263,269,283,293,301,309,320,331,338,347,358,367,376,386,394,405,415,423,435,445,454,463,471,481,491,502,511,521,531,537,549,559,569,580,592,603,613,621,631,639,653,659,669,679,689,699,709,719,729,743,751,758,769,779,793,802,812,821,831,842,851,862,872,883,892,909,917,928,934,943,953,965,974,987,997,1009,1016,1028,1036,1048,1059,1066,1079,1087,1095,1107,1114,1123,1135,1145,1154,1163,1174,1186,1193,1203,1214,1227,1237,1247,1255,1263,1277,1285,1294,1303,1313,1322,1339,1348,1357,1367,1379,1385,1399,1406,1417,1427,1436,

k=6: 7,13,20,29,36,45,53,62,71,81,89,97,109,118,128,137,149,157,167,177,187,197,205,215,228,239,250,258,271,279,292,303,311,326,335,345,358,366,375,386,397,408,418,428,443,453,463,474,488,502,510,521,529,543,554,564,573,587,599,608,619,631,642,655,664,677,691,699,709,723,733,743,755,764,778,790,804,815,825,838,849,862,874,887,902,913,921,934,947,961,969,984,997,1010,1018,1030,1039,1049,1061,1077,1089,1101,1111,1121,1132,1142,1151,1163,1175,1186,1203,1214,1228,1241,1252,1264,1271,1283,1293,1306,1317,1327,1345,1357,1371,1383,1394,1403,1412,1429,1441,1452,1462,1473,1487,1499,1514,1524,1535,1546,1559,1572,1583,1598,1609,1618,1628,1646,1658,1671,

k=7: 8,15,23,33,41,51,59,70,79,89,100,110,121,130,142,151,163,173,185,193,203,215,226,238,249,259,269,279,291,305,314,325,335,343,358,367,379,388,401,412,423,436,446,461,471,482,497,508,517,530,543,554,565,577,588,599,613,625,636,646,658,669,681,691,703,719,729,741,751,763,775,788,798,813,823,835,844,859,876,884,898,909,921,933,941,954,963,977,988,1001,1011,1022,1039,1047,1059,1077,1087,1098,1113,1124,1135,1147,1162,1172,1182,1195,1206,1222,1233,1244,1257,1267,1279,1289,1302,1314,1323,1337,1349,1366,1375,1389,1399,1411,1427,1439,1450,1461,1473,1486,1498,1511,1529,1539,1553,1566,1578,1591,1606,1618,1631,1642,1653,1667,1678,1693,1706,1719,1732,1747,

k=8: 9,17,26,37,46,57,66,77,88,98,110,122,133,145,157,166,177,191,201,212,224,237,248,258,273,283,299,309,320,334,345,356,371,381,395,409,419,430,443,454,466,479,491,505,517,530,542,556,568,582,593,605,615,631,642,654,669,681,692,705,717,730,746,758,769,781,795,808,821,831,843,861,872,887,899,913,923,938,949,964,977,991,1006,1021,1030,1045,1055,1070,1082,1095,1109,1124,1135,1151,1166,1177,1191,1201,1212,1225,1238,1252,1266,1279,1293,1304,1317,1331,1348,1358,1373,1385,1396,1409,1423,1439,1453,1468,1483,1498,1507,1523,1535,1549,1561,1574,1587,1603,1618,1630,1641,1654,1671,1682,1695,1711,1724,1741,1754,1765,1784,1796,1808,1823,1835,1847,1860,1873,1883,1900,

k=9: 10,19,29,41,51,63,73,85,97,108,121,134,144,158,169,182,193,207,221,233,244,258,272,282,297,309,325,336,349,362,376,391,403,416,428,445,457,471,483,499,514,526,539,553,565,578,593,606,623,635,647,661,675,688,705,716,733,745,758,769,786,796,811,825,838,853,868,883,896,908,922,937,954,967,983,996,1009,1023,1038,1053,1063,1081,1097,1112,1124,1137,1147,1163,1177,1193,1204,1219,1231,1245,1259,1277,1288,1302,1316,1329,1343,1359,1373,1385,1402,1415,1429,1442,1459,1473,1488,1499,1514,1527,1543,1559,1574,1589,1604,1619,1636,1647,1662,1674,1689,1705,1719,1733,1753,1765,1781,1795,1809,1823,1838,1852,1868,1888,1901,1915,1926,1945,1959,1975,1987,2004,2019,2032,2045,2062,

k=10: 11,21,32,45,56,69,80,93,106,118,131,145,157,170,182,196,208,222,235,248,261,277,292,304,317,331,344,358,371,386,401,413,428,442,456,468,485,497,509,523,537,553,568,581,597,610,622,639,652,665,679,694,706,721,734,750,762,776,793,807,820,835,851,866,879,894,909,923,938,956,971,983,998,1010,1026,1041,1057,1072,1085,1098,1117,1130,1143,1156,1172,1191,1206,1223,1234,1250,1263,1279,1294,1308,1325,1339,1352,1367,1383,1398,1415,1427,1443,1458,1473,1486,1499,1513,1526,1543,1557,1574,1591,1605,1619,1633,1647,1666,1679,1691,1709,1724,1741,1754,1772,1784,1803,1816,1832,1847,1862,1878,1895,1909,1924,1941,1955,1971,1983,2003,2018,2032,2050,2063,2078,2094,2107,2125,2137,2153,

k=11: 12,23,35,49,61,75,87,101,114,128,142,157,170,186,199,214,227,241,257,270,283,297,313,328,341,357,371,386,401,415,430,446,462,475,491,503,521,536,550,565,581,593,613,627,645,658,674,692,705,722,737,751,764,779,799,815,829,843,861,877,890,906,923,938,955,970,987,1002,1018,1038,1052,1065,1083,1096,1112,1129,1146,1163,1177,1191,1207,1222,1237,1251,1267,1286,1301,1319,1335,1351,1369,1382,1397,1417,1433,1448,1461,1478,1494,1511,1529,1546,1559,1580,1597,1612,1626,1641,1659,1675,1690,1709,1730,1742,1758,1772,1790,1806,1823,1838,1853,1868,1887,1902,1918,1937,1953,1969,1984,1999,2017,2034,2053,2068,2085,2098,2113,2128,2146,2161,2179,2195,2213,2228,2243,2263,2281,2293,2308,2327,

k=12: 13,25,38,53,66,81,94,109,123,138,153,168,183,199,214,230,244,259,278,292,308,322,340,355,370,387,404,420,437,451,469,485,502,516,535,551,568,586,602,619,633,652,668,685,699,714,733,749,768,784,802,817,833,852,867,885,899,917,934,951,965,983,999,1017,1035,1049,1068,1084,1099,1120,1138,1153,1169,1187,1205,1226,1243,1258,1276,1293,1310,1328,1343,1359,1376,1395,1413,1429,1445,1463,1481,1496,1514,1532,1552,1567,1582,1602,1619,1633,1652,1669,1686,1705,1721,1737,1756,1771,1787,1805,1821,1839,1860,1878,1892,1909,1923,1945,1964,1982,1996,2017,2032,2049,2066,2085,2103,2117,2135,2152,2169,2190,2209,2228,2243,2263,2281,2296,2313,2330,2357,2370,2389,2407,2422,2441,2461,2477,2494,2515,

k=13: 14,27,41,57,71,87,101,117,132,148,163,180,195,212,229,245,260,277,295,312,329,344,362,377,393,412,428,445,464,479,496,512,530,547,562,579,597,613,632,649,666,681,701,718,736,753,768,788,803,822,839,858,875,893,911,930,947,963,982,1000,1015,1033,1049,1069,1085,1101,1118,1135,1153,1169,1187,1206,1223,1241,1258,1277,1294,1317,1334,1351,1369,1388,1406,1421,1441,1459,1477,1497,1514,1531,1547,1564,1581,1600,1617,1635,1657,1672,1690,1707,1729,1745,1760,1781,1799,1817,1838,1855,1873,1891,1907,1929,1948,1966,1983,2004,2019,2035,2053,2074,2092,2110,2129,2149,2165,2183,2200,2221,2239,2258,2276,2292,2311,2329,2347,2365,2384,2400,2418,2435,2454,2471,2490,2509,2530,2549,2567,2584,2602,2620,

k=14: 15,29,44,61,76,93,108,125,141,158,173,192,208,226,243,260,277,295,313,332,349,365,385,401,418,437,454,471,490,507,526,543,562,581,597,617,636,654,672,689,709,724,743,763,781,800,817,834,853,872,890,907,925,943,961,979,999,1018,1034,1057,1074,1090,1111,1129,1147,1166,1182,1201,1222,1239,1259,1279,1297,1316,1332,1351,1371,1390,1411,1430,1446,1468,1487,1504,1523,1543,1561,1583,1603,1619,1640,1657,1678,1697,1717,1734,1756,1774,1790,1809,1831,1849,1867,1887,1909,1927,1946,1967,1985,2004,2022,2043,2062,2083,2102,2121,2141,2159,2176,2195,2214,2234,2253,2275,2293,2311,2329,2346,2365,2384,2403,2423,2441,2461,2480,2503,2524,2541,2561,2577,2598,2615,2635,2651,2671,2691,2710,2731,2748,2766,

k=15: 16,31,47,65,81,99,115,133,150,168,184,204,221,239,258,276,294,313,332,350,366,385,406,424,441,461,481,499,518,537,554,573,595,614,632,649,671,689,709,728,747,764,785,804,824,843,861,878,901,918,938,957,978,997,1014,1033,1055,1073,1093,1112,1130,1149,1167,1189,1208,1230,1249,1267,1286,1308,1327,1347,1367,1388,1406,1425,1446,1466,1483,1505,1523,1542,1564,1583,1602,1623,1641,1663,1682,1703,1721,1742,1762,1781,1803,1821,1843,1864,1884,1903,1923,1943,1965,1983,2005,2026,2045,2063,2087,2104,2125,2143,2165,2183,2207,2227,2246,2266,2285,2308,2328,2346,2366,2386,2408,2427,2447,2468,2488,2508,2526,2551,2568,2591,2611,2628,2650,2671,2692,2713,2732,2749,2774,2792,2810,2831,2854,2878,2898,2915,

k=16: 17,33,50,69,86,105,122,141,159,178,195,216,234,253,273,292,311,331,351,371,388,409,431,449,469,489,509,530,549,569,588,608,631,651,670,687,710,729,751,769,789,809,830,849,872,892,912,930,955,974,997,1016,1038,1058,1077,1097,1119,1137,1158,1179,1197,1217,1237,1261,1282,1305,1325,1345,1366,1387,1406,1429,1447,1472,1491,1510,1531,1553,1572,1595,1614,1635,1658,1676,1695,1715,1738,1760,1779,1803,1822,1842,1864,1885,1907,1928,1951,1973,1996,2017,2037,2059,2082,2099,2122,2143,2165,2185,2207,2228,2248,2268,2288,2306,2332,2355,2374,2393,2416,2436,2459,2478,2502,2521,2546,2568,2588,2611,2632,2653,2672,2697,2717,2738,2761,2780,2801,2824,2846,2869,2887,2909,2931,2953,2974,2995,3016,3038,3063,3082,

k=17: 18,35,53,73,91,111,129,149,168,188,206,228,247,267,287,308,328,348,369,391,409,430,452,472,492,512,533,553,575,598,617,638,659,679,699,718,740,762,783,804,826,845,867,887,911,931,953,971,993,1014,1037,1057,1078,1100,1123,1143,1165,1189,1210,1233,1255,1275,1297,1317,1339,1361,1381,1403,1424,1447,1467,1489,1510,1529,1553,1573,1593,1616,1634,1661,1683,1703,1724,1747,1768,1790,1812,1834,1857,1877,1899,1920,1941,1964,1987,2007,2027,2051,2076,2096,2119,2141,2163,2183,2206,2228,2249,2272,2293,2314,2335,2359,2381,2402,2424,2446,2474,2496,2516,2536,2562,2581,2604,2622,2647,2669,2690,2713,2738,2757,2779,2801,2823,2846,2869,2892,2914,2935,2955,2981,3002,3026,3044,3065,3088,3110,3134,3154,3175,3199,

k=18: 19,37,56,77,96,117,136,157,177,198,217,240,260,281,302,324,345,366,388,411,430,452,474,496,516,538,561,582,605,625,647,669,691,713,735,755,778,802,825,846,870,891,914,935,958,980,1003,1023,1047,1069,1093,1115,1137,1160,1182,1204,1227,1251,1272,1298,1320,1341,1363,1385,1409,1432,1453,1475,1497,1522,1543,1567,1589,1611,1637,1659,1679,1704,1724,1751,1774,1796,1819,1841,1863,1886,1908,1931,1955,1978,2001,2024,2047,2068,2093,2116,2136,2162,2188,2211,2232,2255,2279,2302,2325,2348,2372,2393,2419,2439,2462,2487,2510,2533,2556,2578,2605,2626,2648,2670,2696,2721,2743,2765,2788,2811,2835,2858,2882,2902,2929,2949,2972,2993,3017,3043,3063,3087,3111,3137,3161,3184,3204,3227,3252,3275,3301,3324,3346,3369,

k=19: 20,39,59,81,101,123,143,165,186,208,228,252,272,294,317,339,362,384,407,430,450,473,496,519,541,564,587,610,632,655,676,698,723,748,771,791,813,839,862,883,906,928,953,974,998,1025,1046,1068,1091,1114,1139,1163,1186,1206,1231,1253,1275,1299,1323,1344,1371,1392,1415,1438,1464,1490,1511,1535,1559,1581,1606,1630,1654,1676,1702,1727,1747,1772,1793,1817,1843,1867,1889,1915,1936,1961,1983,2008,2031,2056,2081,2104,2127,2149,2173,2198,2221,2247,2271,2293,2317,2341,2364,2388,2408,2434,2458,2482,2503,2529,2554,2575,2602,2624,2645,2669,2692,2719,2744,2765,2789,2813,2839,2864,2885,2909,2936,2958,2982,3006,3031,3054,3079,3102,3124,3149,3174,3196,3219,3243,3269,3293,3317,3341,3367,3388,3412,3436,3461,3486,

k=20: 21,41,62,85,106,129,150,173,195,218,239,264,285,308,332,355,379,402,426,449,472,495,520,544,566,591,614,639,663,687,711,733,760,785,809,831,856,880,905,926,951,974,1000,1022,1047,1074,1097,1121,1145,1169,1196,1221,1246,1267,1293,1316,1340,1367,1391,1413,1442,1464,1490,1513,1539,1566,1589,1614,1639,1664,1689,1716,1739,1765,1788,1815,1836,1861,1883,1910,1937,1961,1986,2012,2034,2060,2085,2109,2136,2161,2187,2212,2235,2261,2285,2311,2335,2362,2387,2413,2440,2467,2491,2515,2538,2564,2589,2612,2638,2664,2689,2713,2738,2762,2786,2810,2837,2865,2889,2913,2939,2964,2992,3015,3041,3064,3091,3117,3144,3169,3193,3217,3245,3267,3294,3319,3344,3370,3394,3422,3447,3470,3496,3519,3544,3569,3595,3622,3645,3673,

k=21: 22,43,65,89,111,135,157,181,204,228,250,276,298,322,347,371,395,420,443,469,493,517,542,568,591,617,641,667,692,717,741,765,793,819,843,867,892,919,943,967,994,1018,1044,1066,1093,1121,1146,1171,1196,1220,1248,1274,1301,1323,1351,1375,1401,1427,1454,1477,1507,1531,1556,1581,1608,1637,1661,1687,1712,1739,1764,1792,1816,1842,1867,1897,1919,1945,1968,1995,2022,2050,2074,2102,2126,2153,2179,2204,2233,2259,2285,2309,2335,2362,2389,2415,2439,2467,2493,2519,2546,2572,2603,2626,2649,2680,2705,2732,2756,2785,2809,2834,2862,2887,2913,2938,2965,2994,3019,3042,3069,3093,3122,3149,3174,3200,3228,3253,3282,3307,3333,3358,3387,3412,3436,3465,3492,3517,3541,3571,3598,3624,3651,3679,3704,3728,3756,3783,3808,3834,

k=22: 23,45,68,93,116,141,164,189,213,238,261,288,311,336,362,387,412,438,462,488,513,537,563,591,615,641,666,693,719,745,771,796,823,849,877,902,926,954,978,1004,1031,1055,1082,1107,1135,1163,1188,1213,1240,1266,1294,1321,1347,1372,1398,1422,1450,1474,1503,1528,1557,1583,1607,1635,1661,1688,1718,1743,1772,1796,1822,1850,1877,1901,1928,1954,1983,2007,2033,2059,2085,2111,2137,2166,2193,2219,2245,2273,2298,2330,2357,2383,2409,2435,2464,2492,2517,2544,2572,2598,2626,2654,2680,2710,2734,2759,2787,2814,2840,2866,2894,2919,2947,2975,3002,3030,3053,3083,3112,3138,3163,3190,3218,3245,3274,3301,3326,3352,3380,3411,3438,3464,3493,3520,3548,3574,3602,3630,3658,3683,3711,3739,3766,3790,3820,3847,3873,3898,3926,3954,

k=23: 24,47,71,97,121,147,171,197,222,248,272,300,324,350,377,403,429,455,481,509,535,560,587,615,640,665,692,719,746,772,800,825,851,879,906,932,957,988,1014,1042,1069,1097,1123,1149,1177,1203,1232,1257,1285,1310,1339,1369,1395,1422,1448,1476,1503,1530,1559,1587,1615,1642,1669,1695,1722,1749,1780,1805,1832,1860,1888,1915,1945,1969,2000,2026,2054,2083,2109,2137,2166,2193,2222,2251,2279,2307,2332,2360,2385,2413,2441,2470,2500,2526,2554,2584,2609,2638,2667,2694,2723,2751,2779,2809,2836,2862,2890,2920,2946,2973,3001,3031,3059,3086,3115,3142,3171,3199,3228,3256,3282,3311,3340,3365,3396,3422,3449,3475,3505,3534,3562,3590,3618,3647,3674,3702,3727,3756,3788,3814,3843,3874,3902,3928,3958,3987,4016,4043,4069,4097,

k=24: 25,49,74,101,126,153,178,205,231,258,283,312,337,364,391,419,446,473,499,529,556,582,610,640,665,694,721,749,778,805,834,861,887,916,945,972,999,1028,1056,1085,1114,1142,1169,1197,1226,1254,1284,1310,1340,1366,1395,1427,1454,1482,1509,1538,1569,1595,1624,1653,1681,1710,1738,1765,1794,1823,1854,1881,1908,1937,1966,1993,2026,2051,2081,2109,2138,2167,2194,2221,2252,2281,2311,2342,2372,2398,2427,2456,2482,2513,2542,2571,2602,2629,2656,2688,2715,2745,2773,2803,2832,2862,2890,2922,2951,2978,3008,3040,3066,3094,3126,3155,3183,3211,3240,3268,3296,3323,3356,3386,3414,3445,3472,3504,3533,3562,3590,3618,3647,3677,3709,3735,3765,3795,3821,3851,3880,3911,3943,3970,4001,4030,4060,4088,4118,4147,4178,4205,4234,4262,

k=25: 26,51,77,105,131,159,185,213,240,268,294,324,350,378,406,435,463,491,518,549,577,604,633,664,690,719,748,777,807,835,864,893,921,951,981,1009,1037,1067,1096,1126,1153,1184,1212,1241,1269,1301,1331,1358,1388,1416,1447,1476,1506,1535,1563,1593,1623,1652,1681,1711,1741,1772,1800,1827,1859,1886,1918,1948,1978,2007,2036,2064,2097,2124,2154,2185,2215,2246,2274,2301,2332,2362,2391,2426,2456,2484,2513,2543,2572,2603,2633,2663,2691,2720,2747,2783,2811,2839,2870,2901,2930,2963,2991,3025,3053,3082,3109,3137,3169,3199,3229,3261,3290,3321,3350,3381,3410,3439,3471,3501,3530,3560,3590,3618,3649,3680,3710,3737,3765,3795,3828,3859,3887,3917,3947,3974,4005,4034,4066,4098,4125,4156,4188,4217,4246,4276,4306,4338,4366,4398,

k=26: 27,53,80,109,136,165,192,221,249,278,305,336,363,392,421,451,479,509,537,569,598,626,656,687,715,745,775,805,835,865,895,925,953,984,1015,1044,1072,1103,1135,1165,1194,1227,1256,1286,1315,1346,1376,1406,1435,1465,1495,1524,1558,1586,1617,1648,1678,1709,1737,1769,1798,1828,1860,1888,1921,1949,1981,2011,2043,2073,2102,2132,2164,2195,2225,2256,2286,2317,2348,2377,2410,2441,2470,2503,2535,2567,2598,2627,2658,2686,2717,2750,2782,2813,2841,2875,2903,2933,2963,2995,3023,3057,3087,3117,3150,3181,3209,3243,3272,3303,3332,3366,3396,3427,3458,3490,3519,3550,3581,3616,3644,3676,3708,3739,3772,3802,3835,3865,3895,3926,3956,3990,4019,4053,4085,4113,4145,4177,4208,4242,4272,4302,4334,4365,4396,4429,4459,4492,4522,4550,

k=27: 28,55,83,113,141,171,199,229,258,288,316,348,376,406,436,467,496,527,556,589,619,648,679,711,740,771,802,832,862,893,925,956,984,1018,1050,1080,1109,1140,1172,1203,1234,1267,1298,1329,1359,1390,1424,1454,1486,1517,1547,1578,1613,1641,1673,1703,1735,1769,1799,1831,1863,1895,1928,1957,1991,2020,2054,2085,2117,2148,2178,2209,2243,2274,2305,2337,2368,2400,2431,2459,2494,2524,2556,2590,2622,2652,2683,2713,2745,2776,2809,2841,2874,2903,2935,2969,3003,3033,3064,3098,3127,3161,3194,3224,3258,3289,3318,3351,3381,3417,3447,3481,3512,3545,3577,3608,3639,3673,3705,3738,3770,3803,3835,3868,3900,3932,3964,3997,4027,4056,4092,4124,4156,4190,4220,4252,4283,4317,4348,4382,4414,4445,4481,4513,4543,4575,4605,4639,4671,4702,

k=28: 29,57,86,117,146,177,206,237,267,298,327,360,389,420,451,483,512,545,575,608,640,670,701,735,765,796,829,861,892,924,956,989,1018,1052,1086,1116,1147,1180,1213,1245,1277,1310,1343,1374,1406,1438,1473,1504,1535,1569,1601,1632,1667,1697,1730,1761,1794,1828,1859,1891,1925,1958,1990,2021,2054,2085,2121,2154,2187,2218,2250,2282,2316,2351,2382,2417,2448,2481,2514,2544,2577,2609,2643,2675,2712,2744,2777,2809,2841,2874,2909,2944,2976,3006,3039,3076,3109,3141,3173,3209,3239,3274,3307,3339,3374,3405,3435,3469,3500,3537,3570,3605,3637,3670,3702,3736,3768,3803,3837,3871,3904,3938,3970,4003,4036,4068,4102,4136,4167,4198,4233,4268,4301,4337,4369,4402,4434,4468,4501,4535,4569,4603,4637,4667,4703,4736,4769,4804,4838,4870,

k=29: 30,59,89,121,151,183,213,245,276,308,338,372,402,434,466,499,529,563,594,628,661,692,724,759,790,822,856,888,921,954,987,1021,1051,1086,1120,1152,1184,1218,1252,1285,1317,1351,1385,1417,1449,1482,1517,1550,1582,1616,1648,1682,1720,1751,1784,1816,1849,1881,1914,1948,1984,2017,2050,2083,2115,2148,2182,2216,2251,2284,2318,2350,2385,2418,2452,2486,2521,2553,2589,2621,2654,2687,2722,2753,2792,2825,2858,2890,2924,2957,2991,3024,3061,3094,3128,3162,3198,3230,3264,3299,3334,3367,3401,3435,3468,3505,3536,3571,3603,3638,3671,3706,3740,3774,3807,3844,3877,3910,3944,3980,4015,4048,4083,4118,4149,4185,4219,4254,4286,4319,4353,4388,4423,4457,4493,4525,4559,4592,4626,4662,4695,4729,4765,4797,4829,4866,4901,4934,4970,5004,

k=30: 31,61,92,125,156,189,220,253,285,318,349,384,415,448,481,515,546,581,613,648,681,714,747,783,815,848,883,916,950,984,1017,1051,1082,1118,1153,1186,1218,1254,1289,1322,1356,1390,1424,1459,1492,1528,1563,1598,1631,1666,1698,1733,1773,1805,1838,1871,1906,1941,1975,2009,2045,2081,2114,2149,2181,2216,2252,2287,2322,2357,2391,2425,2462,2495,2531,2564,2600,2633,2671,2704,2738,2773,2809,2841,2879,2912,2947,2980,3015,3048,3085,3118,3158,3191,3226,3263,3297,3331,3365,3400,3437,3471,3509,3544,3578,3615,3647,3684,3716,3754,3788,3824,3859,3894,3928,3963,3998,4034,4069,4104,4138,4175,4210,4246,4282,4317,4353,4388,4423,4457,4492,4529,4564,4601,4638,4671,4707,4740,4777,4813,4847,4883,4919,4954,4987,5025,5056,5092,5127,5164,

Best,

É.

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(April 19th, 2007: some big files computed by Sydney Roussel -- 20000 terms for k = 2 to 5 -- are here)

Please find here the .pdf paper of Jean-Paul Delahaye about this very subject, which was published in march 2007 in the french magazine « Pour la Science ».