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A143773 Number of partitions of n such that every part is divisible by number of parts.

%I A143773 #22 Mar 03 2025 13:31:02
%S A143773 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 A143773 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 A143773 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 A143773 Number of partitions of n such that every part is divisible by number of parts.
%H A143773 Alois P. Heinz, <a href="/A143773/b143773.txt">Table of n, a(n) for n = 1..10000</a>
%F A143773 G.f.: Sum(x^(k^2)/Product(1-x^(k*i), i=1..k), k=1..infinity).
%F A143773 For prime p, a(p) = 1 and a(p^2) = 2. For odd prime p, a(2*p) = (p + 1)/2. - _Peter Bala_, Mar 03 2025
%e A143773 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 A143773 m = 100;
%t A143773 gf = Sum[x^(k^2)/Product[1-x^(k*i), {i, 1, k}], {k, 1, Sqrt[m]//Ceiling}];
%t A143773 CoefficientList[gf + O[x]^m, x] // Rest (* _Jean-François Alcover_, May 13 2019 *)
%o A143773 (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 A143773 Cf. A000005, A000041, A067538, A143862, A298422, A298423, A298426.
%K A143773 easy,nonn,look
%O A143773 1,4
%A A143773 _Vladeta Jovovic_, Aug 31 2008
%E A143773 More terms from _Max Alekseyev_, May 03 2009