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%I #25 Nov 11 2015 12:14:56
%S 1,1,1,2,3,6,4,14,11,19,16,55,20,100,50,72,96,296,79,489,167,328,349,
%T 1254,271,1331,816,1435,980,4564,506,6841,2745,4027,3840,6816,2366,
%U 21636,7845,12027,6583,44582,4293,63260,18700,20259,29920,124753,15842,120351
%N Number of partitions of n such that no part is a prime divisor of n.
%C If n is prime, a(n) = A000041(n) - 1.
%H Alois P. Heinz, <a href="/A237125/b237125.txt">Table of n, a(n) for n = 0..1000</a>
%e With n = 6, the prime divisors of 6 are 2 and 3, and the partitions of 6 that do not contain either 2 or 3 are: [6], [5,1], [4,1,1] and [1,1,1,1,1,1], so a(6) = 4.
%p with(numtheory): with(combinat):
%p a:= proc(n) local b, s; s:= factorset(n);
%p b:= proc(n, i) option remember;
%p `if`(n=0, 1, `if`(i<1, 0, b(n, i-1)+
%p `if`(i>n or i in s, 0, b(n-i, i))))
%p end;
%p `if`(isprime(n), numbpart(n)-1, b(n$2))
%p end:
%p seq(a(n), n=0..50); # _Alois P. Heinz_, Feb 04 2014
%t a[1]=1; a[n_] := If[PrimeQ[n], PartitionsP[n] - 1, Block[{p = First /@ FactorInteger@n}, Length@ Select[ IntegerPartitions[n], Intersection[#, p] == {} &]]]; Array[a, 50] (* _Giovanni Resta_, Feb 04 2014 *)
%t a[0] = a[1] = 1; a[n_] := Module[{b, s}, s = FactorInteger[n][[All,1]]; b[m_, i_] := b[m, i] = If[m == 0, 1, If[i<1, 0, b[m, i-1]+ If[i>m || MemberQ[s, i], 0, b[m-i, i]]]]; If[PrimeQ[n], PartitionsP[n]-1, b[n, n] ]]; Table[a[n], {n, 0, 50}] (* _Jean-François Alcover_, Nov 11 2015, after _Alois P. Heinz_ *)
%Y Cf. A236937, A236938.
%K nonn,look
%O 0,4
%A _J. Stauduhar_, Feb 03 2014
%E a(8)-a(49) from _Giovanni Resta_, Feb 04 2014
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