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A088044 Number of partitions of n with distinct prime signatures.

%I A088044 #14 Sep 25 2023 04:58:08
%S A088044 1,1,2,2,2,3,4,4,5,6,8,10,10,11,14,14,19,22,22,25,30,34,38,40,46,54,
%T A088044 60,63,74,81,85,91,108,118,126,134,153,173,180,186,214,236,245,262,
%U A088044 301,326,338,355,395,437,461,476,540,592,610,649,717,768,811,853,943,1039,1078
%N A088044 Number of partitions of n with distinct prime signatures.
%C A088044 Essentially the same as A077564. [From _R. J. Mathar_, Sep 11 2008]
%e A088044 a(9) = 6 and the partitions are (9), (1,8), (3,6),(4,5), (1,2,6),(2,3,4).
%e A088044 The partitions (2,7), (1,3,5),(1,4,4), etc. are not considered for the count as 2 and 7 both are prime and so on.
%o A088044 (PARI) ps(n) = local(f); f = factor(n); vecsort(f[,2]);
%o A088044 b(n, lim, used, cUsed) = local(s, v, found, j, used2); if (n == 0, return(1)); if (n > lim*(lim + 1)/2, return(0)); s = 0; for (i = 1, lim, v = ps(i); found = 0; j = 1; while (!found && j <= cUsed, if (v == used[j], found = 1, j++)); if (!found, used2 = used; used2[1 + cUsed] = v; s += b(n - i, min (n - i, i - 1), used2, cUsed + 1))); s;
%o A088044 a(n) = b(n, n, vector(12), 0); \\ _David Wasserman_, Jun 21 2005
%Y A088044 Cf. A088045.
%K A088044 nonn
%O A088044 1,3
%A A088044 _Amarnath Murthy_, Sep 20 2003
%E A088044 More terms from _David Wasserman_, Jun 21 2005