%I #16 May 21 2021 08:11:33
%S 1,1,1,2,1,4,1,4,4,5,1,15,1,7,14,17,1,28,1,40,28,11,1,99,31,13,49,99,
%T 1,186,1,152,76,17,208,425,1,19,109,699,1,584,1,433,823,23,1,1625,437,
%U 1140,193,746,1,2003,1748,2749,244,29,1,7404,1,31,4158,3258,3766,6307,1
%N Number of partitions of n into distinct parts in which the number of parts divides n.
%H Alois P. Heinz, <a href="/A102627/b102627.txt">Table of n, a(n) for n = 1..1000</a>
%e From _Gus Wiseman_, Sep 24 2019: (Start)
%e The a(1) = 1 through a(12) = 15 strict integer partitions whose average is an integer (A = 10, B = 11, C = 12):
%e (1) (2) (3) (4) (5) (6) (7) (8) (9) (A) (B) (C)
%e (31) (42) (53) (432) (64) (75)
%e (51) (62) (531) (73) (84)
%e (321) (71) (621) (82) (93)
%e (91) (A2)
%e (B1)
%e (543)
%e (642)
%e (651)
%e (732)
%e (741)
%e (831)
%e (921)
%e (5421)
%e (6321)
%e (End)
%p a:= proc(m) option remember; local b; b:=
%p proc(n, i, t) option remember; `if`(i*(i+1)/2<n,
%p 0, `if`(n=0, `if`(irem(m, t)=0, 1, 0),
%p b(n, i-1, t)+b(n-i, min(n-i, i-1), t+1)))
%p end: `if`(isprime(m), 1, b(m$2, 0))
%p end:
%p seq(a(n), n=1..100); # _Alois P. Heinz_, Sep 25 2019
%t npdp[n_]:=Count[Select[IntegerPartitions[n],Length[#]==Length[ Union[ #]]&], _?(Divisible[n,Length[#]]&)]; Array[npdp,70] (* _Harvey P. Dale_, Feb 12 2016 *)
%t a[m_] := a[m] = Module[{b}, b[n_, i_, t_] := b[n, i, t] = If[i(i+1)/2 < n, 0, If[n == 0, If[Mod[m, t] == 0, 1, 0], b[n, i - 1, t] + b[n - i, Min[n - i, i - 1], t + 1]]]; If[PrimeQ[m], 1, b[m, m, 0]]];
%t Array[a, 100] (* _Jean-François Alcover_, May 21 2021, after _Alois P. Heinz_ *)
%Y The BI-numbers of these partitions are given by A326669 (numbers whose binary indices have integer mean).
%Y The non-strict case is A067538.
%Y Strict partitions with integer geometric mean are A326625.
%Y Strict partitions whose maximum divides their sum are A326850.
%Y Cf. A018818, A033630, A316413, A326622, A326843, A326851.
%K easy,nonn
%O 1,4
%A _Vladeta Jovovic_, Feb 01 2005