cp's OEIS Frontend

A235945 Number of partitions of n containing at least one prime.

%I A235945 #15 Jan 28 2014 10:39:08
%S A235945 0,0,1,2,3,5,8,12,17,24,34,48,65,88,118,157,205,269,348,450,575,734,
%T A235945 929,1176,1473,1845,2297,2856,3527,4355,5346,6558,8004,9759,11848,
%U A235945 14374,17363,20958,25210,30292,36278,43412,51792,61733,73383,87146,103239,122194
%N A235945 Number of partitions of n containing at least one prime.
%H A235945 Alois P. Heinz, <a href="/A235945/b235945.txt">Table of n, a(n) for n = 0..1000</a>
%F A235945 a(n) = A000041(n) - A002095(n).
%F A235945 Product_{k>0} 1/(1-x^k) - Product_{k>0} (1-x^prime(k))/(1-x^k). - _Alois P. Heinz_, Jan 18 2014
%e A235945 a(5) = 5 because 5 partitions of 5 contain at least one prime: [5], [3,2], [3,1,1], [2,2,1], [2,1,1,1].
%p A235945 b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0,
%p A235945        b(n, i-1)+`if`(i>n or isprime(i), 0, b(n-i, i))))
%p A235945     end:
%p A235945 a:= n-> combinat[numbpart](n) -b(n, n):
%p A235945 seq(a(n), n=0..50);  # _Alois P. Heinz_, Jan 18 2014
%t A235945 b[n_, i_] := b[n, i] = If[n == 0, 1, If[i<1, 0, b[n, i-1] + If[i>n || PrimeQ[i], 0, b[n-i, i]]]]; a[n_] := PartitionsP[n]-b[n, n]; Table[a[n], {n, 0, 50}] (* _Jean-François Alcover_, Jan 28 2014, after _Alois P. Heinz_ *)
%Y A235945 Cf. A000041, A002095.
%K A235945 nonn
%O A235945 0,4
%A A235945 _J. Stauduhar_, Jan 17 2014