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%I #15 Sep 09 2019 10:24:37
%S 1,1,2,3,3,5,4,9,6,11,5,37,6,21,22,33,7,88,8,117,36,37,9,533,12,47,48,
%T 200,10,898,11,306,68,69,70,2124,12,82,83,1864,13,2298,14,612,613,109,
%U 15,14892,19,755,125,771,16,4669,141,4810,142,143,17,120808,18,177,1265
%N a(n) = number of partitions of d(n) into d(k)'s, where the k's are each <= n and distinct, but the d(k)'s need not be distinct. Here d(m) = the number of divisors of m.
%C a(n) = the coefficient of x^d(n) in the product: Product_{k=1..n} (1 + x^d(k)).
%e The first 10 terms of the number-of-divisors sequence (A000005) are 1,2,2,3,2,4,2,4,3,4. For a(10), we want the number of partitions of these 10 terms that sum to d(10)=4. We have: d(10)=4, d(8)=4, d(6)=4, d(9)+d(1) = 3+1, d(4)+d(1) = 3+1, d(7)+d(5) = d(7)+d(3) = d(7)+d(2) = d(5)+d(3) = d(5)+d(2) = d(3)+d(2) = 2+2. So there are 11 such sums (including those equal to 4 itself). Therefore a(10)=11.
%p A175108 := proc(n) g := 1 ; for k from 1 to n do g := g*(1+x^numtheory[tau](k)) ; g := expand(g) ; end do ; coeftayl(g,x=0,numtheory[tau](n)) ; end proc: seq(A175108(n),n=1..80) ; # _R. J. Mathar_, Mar 05 2010
%Y Cf. A000005.
%K nonn
%O 1,3
%A _Leroy Quet_, Feb 12 2010
%E More terms from _R. J. Mathar_ and _Ray Chandler_, Mar 05 2010
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