A362631 Lexicographically earliest infinite sequence of distinct positive integers with a(n) = n for n <= 3, and for n > 3 a(n) is the least novel multiple of the greatest prime divisor of a(n-2) which does not divide a(n-1).
1, 2, 3, 4, 6, 5, 9, 10, 12, 15, 8, 20, 7, 25, 14, 30, 21, 35, 18, 28, 24, 42, 11, 49, 22, 56, 33, 63, 44, 70, 55, 77, 40, 66, 45, 88, 50, 99, 60, 110, 27, 121, 36, 132, 13, 143, 16, 26, 17, 39, 34, 52, 51, 65, 68, 78, 85, 91, 102, 104, 119, 117, 136, 130, 153, 156, 170, 169, 187, 182, 204, 195, 221, 75, 238, 80, 255, 32, 272, 19
Offset: 1
Keywords
Examples
a(4,5) = 4, 6 and since rad(4)|rad(6) a(6) = least u = 5. a(11,12) = 8, 20 and since rad(8)|rad(20) a(13) = least u = 7. a(44,45) = 132, 13 and gpd(132) = 11 does not divide 13, and since it is the 13th occurrence of p = 11, a(46) = 13*11 = 143. a(45,46) = 13, 143 which forces a(47) = least u = 16 (see Comment). a(90,91) = 69, 114 and 23 is the greatest prime dividing 69 which does not divide 114. Since 23 has not appeared earlier in the sequence a(92) = 23. There is as yet no known formula for the row lengths of the table below. Whereas most rows terminate with a multiple of the prime they start with, there are exceptions, e.g., 47, 109. This behavior is open to explanation. The table starts: 2; 3,4,6; 5,9,10,12,15,8,20; 7,25,14,30,21,35,18,28,24,42; 11,49,22,56,33,63,44,70,55,77,40,66,45,88,50,99,60,110,27,121,36,132; 13,143,16,26; 17,39,34,52,51,65,68,78,85,91,102,104,119,117,136,130,153,156,170,169,187,182,204,195,221,75,238,80,255,32,272; 19,....
Links
- Michael De Vlieger, Table of n, a(n) for n = 1..10000
- Michael De Vlieger, Scatterplot of a(n) n = 1..10000.
- Michael De Vlieger, Log log scatterplot of a(n) n = 1..2^20.
- Michael De Vlieger, Log log scatterplot of a(n), n = 1..10000, showing primes in red, composite prime powers in gold, squarefree composites in green, and numbers neither squarefree nor prime powers in blue. We accentuate powerful numbers that are not prime powers in light blue.
Programs
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Mathematica
mm = 3; nn = 10^4; c[] := False; q[] := 1; Array[Set[{a[#], c[#]}, {#, True}] &, mm]; q[2]++; u = mm + 1; i = a[mm - 1]; j = a[mm]; s = {1}; Do[t = FactorInteger[i][[All, 1]]; If[SubsetQ[s, t], q[u]++; Set[k, u], p = SelectFirst[Reverse@ t, CoprimeQ[j, #] &]; If[! IntegerQ[p], Set[k, u], While[c[p*q[p]], q[p]++]; k = p*q[p]] ]; Set[{a[n], c[k], i, j, s}, {k, True, j, k, t}]; If[k == u, While[c[u], u++]], {n, mm + 1, nn}]; Array[a, nn]
Comments