A363956 a(1) = 1, a(2) = 2; for n > 2, a(n) is the smallest positive number that has not yet appeared that is a multiple of prime(omega(a(n-1))).

1, 2, 4, 6, 3, 8, 10, 9, 12, 15, 18, 21, 24, 27, 14, 30, 5, 16, 20, 33, 36, 39, 42, 25, 22, 45, 48, 51, 54, 57, 60, 35, 63, 66, 40, 69, 72, 75, 78, 50, 81, 26, 84, 55, 87, 90, 65, 93, 96, 99, 102, 70, 80, 105, 85, 108, 111, 114, 95, 117, 120, 100, 123, 126, 110, 115, 129, 132, 125, 28, 135, 138

Offset

1,2

Comments

The sequence is conjectured to be a permutation of the positive integers, although it takes many terms for the primes to appear, e.g., a(210667) = 17. After the first two terms the only fixed points in the first 500000 terms are 15777 and 86977, although it is possible more exist.

Links

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

Scott R. Shannon, Image of the first 500000 terms. The green line is a(n) = n.

Michael De Vlieger, Log log plot of a(n), n = 1..2^20.

Michael De Vlieger, Log log plot of a(n), n = 1..2^12, showing primes in red, composite prime powers in gold, squarefree composites in green, and numbers neither squarefree nor composite in blue. Powerful numbers that are not prime powers are highlighted in light blue.

Example

a(10) = 15 as prime(omega(a(9))) = prime(A001221(12)) = prime(2) = 3, and 15 is the smallest unused number that is a multiple of 3.

Mathematica

nn = 120; c[_] := False; m[_] := 1; f[x_] := Prime@ PrimeNu[x];
Array[Set[{a[#], c[#], m[#]}, {#, True, 2}] &, 2]; j = 2;
Do[k = f[j];
 While[c[m[k] k], m[k]++]; k *= m[k];
 Set[{a[n], c[k], j}, {k, True, k}], {n, 3, nn}];
Array[a, nn] (* Michael De Vlieger, Jul 01 2023 *)

Crossrefs

Cf. A363504, A001221, A000040, A001222, A351495.

Sequence in context: A350033 A351625 A352976 * A026200 A026218 A181473

Adjacent sequences: A363953 A363954 A363955 * A363957 A363958 A363959

Keyword

nonn

Author

Scott R. Shannon, Jun 29 2023

Status

approved