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A085507 Stirling transform of the prime characteristic function.

%I A085507 #19 Feb 16 2025 08:32:50
%S A085507 0,0,1,4,13,41,136,505,2171,10693,58246,340242,2095435,13492077,
%T A085507 90267633,623383765,4414350135,31899350954,235002008725,1773013299342,
%U A085507 13855253098226,114135759054965,1010686200326760,9744658443894282,102153128291263124,1147158516520205256
%N A085507 Stirling transform of the prime characteristic function.
%H A085507 Alois P. Heinz, <a href="/A085507/b085507.txt">Table of n, a(n) for n = 0..576</a>
%H A085507 S. K. Ghosal, J. K. Mandal, <a href="https://doi.org/10.1016/j.protcy.2013.12.341">Stirling Transform Based Color Image Authentication</a>, Procedia Technology, 2013 Volume 10, 2013, Pages 95-104.
%H A085507 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/StirlingTransform.html">Stirling Transform</a>
%F A085507 G.f.: Sum_{k>=1} x^prime(k)/Product_{j=1..prime(k)} (1 - j*x). - _Ilya Gutkovskiy_, Jun 19 2018
%p A085507 b:= proc(n, m) option remember;
%p A085507      `if`(n=0, `if`(isprime(m), 1, 0), m*b(n-1, m)+b(n-1, m+1))
%p A085507     end:
%p A085507 a:= n-> b(n, 0):
%p A085507 seq(a(n), n=0..25);  # _Alois P. Heinz_, Aug 06 2021
%t A085507 a[n_] := Sum[ StirlingS2[n, k]*Boole[PrimeQ[k]], {k, 0, n}]; Table[a[n], {n, 0, 23}] (* _Jean-François Alcover_, Oct 29 2012 *)
%K A085507 nonn
%O A085507 0,4
%A A085507 _Eric W. Weisstein_, Jul 02 2003