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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