A097296 Numbers k such that A001055(k) divides k.

1, 2, 3, 4, 5, 6, 7, 10, 11, 12, 13, 14, 17, 19, 20, 22, 23, 26, 27, 28, 29, 30, 31, 34, 36, 37, 38, 41, 43, 44, 46, 47, 48, 52, 53, 56, 58, 59, 61, 62, 67, 68, 70, 71, 73, 74, 76, 79, 82, 83, 86, 89, 92, 94, 97, 101, 103, 105, 106, 107, 109, 110, 113, 116, 118, 122, 124, 127, 130

Offset: 1

Author

N. J. A. Sloane, Jun 12 2009

Keywords

• nonn

Links

• Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..1000 from Alois P. Heinz)
• Florian Luca, Anirban Mukhopadhyay and Kotyada Srinivas, On the Oppenheim's "factorisatio numerorum" function, arXiv:0807.0986 [math.NT], 2008.

Crossrefs

Cf. A001055.

Programs

Maple

g:= proc(n, k) option remember; `if`(n>k, 0, 1)+
      `if`(isprime(n), 0, add(`if`(d>k, 0, g(n/d, d)),
         d=numtheory[divisors](n) minus {1, n}))
    end:
a:= proc(n) option remember; local k;
      for k from 1+`if`(n=1, 0, a(n-1))
      while irem(k, g(k$2))>0 do od; k
    end:
seq(a(n), n=1..100);  # Alois P. Heinz, May 16 2014

Mathematica

g[n_, k_] := g[n, k] = If[n > k, 0, 1] + If[PrimeQ[n], 0, Sum[If[d > k, 0, g[n/d, d]], {d, Divisors[n] // Most // Rest}]]; a[1] = 1; a[n_] := (For[k = 1 + If[n == 1, 0, a[n-1]], Mod[k, g[k, k]] > 0 , k++]; k); Table[a[n], {n, 1, 100}] (* Jean-François Alcover, Oct 07 2014, after Alois P. Heinz *)

Formula

Luca et al. estimate the density of this sequence (see their Theorem 3).

The number of terms that do not exceed x is ~ x/(log(x))^(1+o(1)) (Luca et al., 2008). - Amiram Eldar, May 23 2024