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A102627
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Number of partitions of n into distinct parts in which the number of parts divides n.
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82
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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
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OFFSET
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1,4
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LINKS
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EXAMPLE
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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)
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MAPLE
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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:
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MATHEMATICA
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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]]];
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CROSSREFS
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The BI-numbers of these partitions are given by A326669 (numbers whose binary indices have integer mean).
Strict partitions with integer geometric mean are A326625.
Strict partitions whose maximum divides their sum are A326850.
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KEYWORD
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easy,nonn
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AUTHOR
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STATUS
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approved
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