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

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, 1, 186, 1, 152, 76, 17, 208, 425, 1, 19, 109, 699, 1, 584, 1, 433, 823, 23, 1, 1625, 437, 1140, 193, 746, 1, 2003, 1748, 2749, 244, 29, 1, 7404, 1, 31, 4158, 3258, 3766, 6307, 1

Offset

1,4

Links

Alois P. Heinz, Table of n, a(n) for n = 1..1000

Example

From Gus Wiseman, Sep 24 2019: (Start)
The a(1) = 1 through a(12) = 15 strict integer partitions whose average is an integer (A = 10, B = 11, C = 12):
  (1)  (2)  (3)  (4)   (5)  (6)    (7)  (8)   (9)    (A)   (B)  (C)
                 (31)       (42)        (53)  (432)  (64)       (75)
                            (51)        (62)  (531)  (73)       (84)
                            (321)       (71)  (621)  (82)       (93)
                                                     (91)       (A2)
                                                                (B1)
                                                                (543)
                                                                (642)
                                                                (651)
                                                                (732)
                                                                (741)
                                                                (831)
                                                                (921)
                                                                (5421)
                                                                (6321)
(End)

Maple

a:= proc(m) option remember; local b; b:=
      proc(n, i, t) option remember; `if`(i*(i+1)/2<n,
         0, `if`(n=0, `if`(irem(m, t)=0, 1, 0),
         b(n, i-1, t)+b(n-i, min(n-i, i-1), t+1)))
      end: `if`(isprime(m), 1, b(m$2, 0))
    end:
seq(a(n), n=1..100);  # Alois P. Heinz, Sep 25 2019

Mathematica

npdp[n_]:=Count[Select[IntegerPartitions[n], Length[#]==Length[ Union[ #]]&], _?(Divisible[n, Length[#]]&)]; Array[npdp, 70] (* Harvey P. Dale, Feb 12 2016 *)
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]]];
Array[a, 100] (* Jean-François Alcover, May 21 2021, after Alois P. Heinz *)

Crossrefs

The BI-numbers of these partitions are given by A326669 (numbers whose binary indices have integer mean).

The non-strict case is A067538.

Strict partitions with integer geometric mean are A326625.

Strict partitions whose maximum divides their sum are A326850.

Cf. A018818, A033630, A316413, A326622, A326843, A326851.

Sequence in context: A079277 A066452 A007104 * A296560 A284652 A261242

Adjacent sequences: A102624 A102625 A102626 * A102628 A102629 A102630

Keyword

easy,nonn

Author

Vladeta Jovovic, Feb 01 2005

Status

approved