cp's OEIS Frontend

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

%I A161993 #24 Nov 20 2020 04:27:04
%S A161993 1,3,8,19,43,85,171,315,580,1022,1766,2982,4959,8081,12997,20596,
%T A161993 32261,49909,76447,115872,174133,259312,383206,561877,818225,1183266,
%U A161993 1700658,2429266,3450562,4874167,6850072,9578548,13331445,18469783,25478494,34999375,47887091
%N A161993 A006005 (shifted) convolved with all of its regularly "aerated" variants.
%C A161993 Refer to A161779 for the analogous sequence based on the factorials.
%C A161993 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*...
%H A161993 Vaclav Kotesovec, <a href="/A161993/b161993.txt">Table of n, a(n) for n = 0..10000</a> (terms 0..5000 from Alois P. Heinz)
%p A161993 p:= n-> `if`(n=0, 1, ithprime(n+1)):
%p A161993 b:= proc(n, i) option remember; `if`(i>n, 0,
%p A161993      `if`(irem(n, i, 'r')=0, p(r), 0)+
%p A161993       add(p(j)*b(n-i*j, i+1), j=0..n/i))
%p A161993     end:
%p A161993 a:= n-> `if`(n=0, 1, b(n, 1)):
%p A161993 seq(a(n), n=0..45);  # _Alois P. Heinz_, Jul 27 2019
%t A161993 p[n_] := If[n==0, 1, Prime[n+1]];
%t A161993 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}]];
%t A161993 a[n_] := If[n==0, 1, b[n, 1]];
%t A161993 a /@ Range[0, 45] (* _Jean-François Alcover_, Nov 20 2020, after _Alois P. Heinz_ *)
%Y A161993 Cf. A006005, A161779.
%K A161993 nonn
%O A161993 0,2
%A A161993 _Gary W. Adamson_, Jun 24 2009
%E A161993 Definition and comment corrected by _Omar E. Pol_, Aug 18 2011
%E A161993 Correct offset and a(13)-a(36) from _Alois P. Heinz_, Jul 27 2019