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A102627 Number of partitions of n into distinct parts in which the number of parts divides n. 82
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 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,4
LINKS
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.
Sequence in context: A079277 A066452 A007104 * A296560 A284652 A261242
KEYWORD
easy,nonn
AUTHOR
Vladeta Jovovic, Feb 01 2005
STATUS
approved

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Last modified March 29 05:16 EDT 2024. Contains 371264 sequences. (Running on oeis4.)