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Number of partitions of n such that every part is divisible by number of parts.
54

%I #19 May 13 2019 02:46:43

%S 1,1,1,2,1,2,1,3,2,3,1,5,1,4,3,6,1,8,1,7,5,6,1,14,2,7,8,11,1,17,1,14,

%T 11,9,3,29,1,10,15,23,1,28,1,23,25,12,1,51,2,20,25,32,1,44,11,39,31,

%U 15,1,94,1,16,40,52,19,64,1,57,45,44,1,126,1,19,83,74,6,90,1,124,63,21,1,186

%N Number of partitions of n such that every part is divisible by number of parts.

%H Alois P. Heinz, <a href="/A143773/b143773.txt">Table of n, a(n) for n = 1..10000</a>

%F G.f.: Sum(x^(k^2)/Product(1-x^(k*i), i=1..k), k=1..infinity).

%e The a(18) = 8 partitions are (18), (10 8), (12 6), (14 4), (16 2), (6 6 6), (9 6 3), (12 3 3). - _Gus Wiseman_, Jan 26 2018

%t m = 100;

%t gf = Sum[x^(k^2)/Product[1-x^(k*i), {i, 1, k}], {k, 1, Sqrt[m]//Ceiling}];

%t CoefficientList[gf + O[x]^m, x] // Rest (* _Jean-François Alcover_, May 13 2019 *)

%o (PARI) Vec(sum(k=1,20,x^(k^2)/prod(i=1,k,1-x^(k*i)+O(x^400)))) \\ _Max Alekseyev_, May 03 2009

%Y Cf. A000005, A000041, A067538, A298422, A298423, A298426.

%K easy,nonn,look

%O 1,4

%A _Vladeta Jovovic_, Aug 31 2008

%E More terms from _Max Alekseyev_, May 03 2009