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A102627 Number of partitions of n into distinct parts in which the number of parts divides n.

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