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

0,2

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)

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 *)

Crossrefs

Cf. A006005, A161779.

Sequence in context: A357291 A099050 A065352 * A360489 A259401 A008466

Adjacent sequences: A161990 A161991 A161992 * A161994 A161995 A161996

Keyword

nonn

Author

Gary W. Adamson, Jun 24 2009

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

Status

approved