Voozh

A382868

a(1) = 1, a(2) = 2. For n > 2 a(n) is the smallest novel number divisible by the smallest prime p which divides a(n-1) but does not divide a(n-2). If no such prime exists a(n) is the least novel k such that gcd(k, a(n-1)) > 1.

1, 2, 4, 6, 3, 9, 12, 8, 10, 5, 15, 18, 14, 7, 21, 24, 16, 20, 25, 30, 22, 11, 33, 27, 36, 26, 13, 39, 42, 28, 32, 34, 17, 51, 45, 35, 49, 56, 38, 19, 57, 48, 40, 50, 44, 55, 60, 46, 23, 69, 54, 52, 65, 70, 58, 29, 87, 63, 77, 66, 62, 31, 93, 72, 64, 68, 85, 75

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OFFSET

1,2

COMMENTS

Similar to A064413, from which it departs at a(19) = 25. Any odd prime p is preceded by a multiple of itself. Conjectured to be a permutation of the natural numbers, with primes occurring in natural order.

LINKS

Michael De Vlieger, Table of n, a(n) for n = 1..10000

Michael De Vlieger, Log log scatterplot of a(n), n = 1..2^16, showing primes in red, proper prime powers in gold, squarefree composites in green, and numbers neither squarefree nor prime powers in blue and purple, where purple additionally represents powerful numbers that are not prime powers. Primorials are highlighted with large green points.

EXAMPLE

a(3) = 4 since 2|2 but 2 !| 1. Then since there is no prime which dives 4 but not 2, a(4) = 6, the least novel k such that gcd(k, 4) > 1. Since 3|6 but 3!|4, a(5) = 3.

Every prime term is followed by the least novel multiple of itself.

MATHEMATICA

nn = 120; c[_] := False; m[_] := 1;

Set[{i, j, s, t, u, c[1], c[2]}, {1, 2, {}, {2}, 3, True, True}];

{1, 2}~Join~Reap[

Do[p = If[Length[#] == 0, 0, First[#]] &@ Complement[t, s];

If[p == 0,

k = u; While[Or[c[k], CoprimeQ[j, k]], k++],

While[c[Set[k, p*m[p]]], m[p]++] ];

Set[{c[k], i, j, s, t}, {True, j, k, t, Reverse@ FactorInteger[k][[All, 1]] } ];

If[k == u, While[c[u], u++]]; Sow[k],

{n, 3, nn}] ][[-1, 1]] (* Michael De Vlieger, Apr 07 2025 *)