A161993 A006005 (shifted) convolved with all of its regularly "aerated" variants.

1, 3, 8, 19, 43, 85, 171, 315, 580, 1022, 1766, 2982, 4959, 8081, 12997, 20596, 32261, 49909, 76447, 115872, 174133, 259312, 383206, 561877, 818225, 1183266, 1700658, 2429266, 3450562, 4874167, 6850072, 9578548, 13331445, 18469783, 25478494, 34999375, 47887091

Offset: 0

Author

Gary W. Adamson, Jun 24 2009

Keywords

• nonn

Comments

Refer to A161779 for the analogous sequence based on the factorials.

Given A006005 (1 together with the odd primes = odd noncomposite numbers) = a, then b = the aerated variant: (1, 0, 3, 0, 5, 0, 7,...); c = (1, 0, 0, 3, 0, 0, 5,...) and so on such that A161993 = the infinite convolution product: a*b*c*...

Links

• Vaclav Kotesovec, Table of n, a(n) for n = 0..10000 (terms 0..5000 from Alois P. Heinz)

Crossrefs

Cf. A006005, A161779.

Programs

Maple

p:= n-> `if`(n=0, 1, ithprime(n+1)):
b:= proc(n, i) option remember; `if`(i>n, 0,
     `if`(irem(n, i, 'r')=0, p(r), 0)+
      add(p(j)*b(n-i*j, i+1), j=0..n/i))
    end:
a:= n-> `if`(n=0, 1, b(n, 1)):
seq(a(n), n=0..45);  # Alois P. Heinz, Jul 27 2019

Mathematica

p[n_] := If[n==0, 1, Prime[n+1]];
b[n_, i_] := b[n, i] = If[i>n, 0, If[Mod[n, i]==0, p[n/i], 0] + Sum[p[j] b[n - i j, i+1], {j, 0, n/i}]];
a[n_] := If[n==0, 1, b[n, 1]];
a /@ Range[0, 45] (* Jean-François Alcover, Nov 20 2020, after Alois P. Heinz *)

Extensions

Definition and comment corrected by Omar E. Pol, Aug 18 2011

Correct offset and a(13)-a(36) from Alois P. Heinz, Jul 27 2019