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A332887
a(n) is the number of partition numbers > 1 that are proper divisors of the n-th partition number.
1
0, 0, 0, 0, 0, 2, 2, 4, 3, 2, 2, 0, 3, 3, 4, 2, 4, 3, 2, 4, 2, 1, 4, 3, 4, 3, 3, 2, 2, 3, 2, 2, 2, 2, 0, 3, 2, 10, 4, 5, 3, 3, 1, 1, 2, 2, 2, 2, 3, 2, 5, 2, 5, 2, 1, 2, 4, 4, 0, 8, 3, 1, 4, 1, 2, 0, 4, 1, 2, 3, 1, 0, 4, 9, 1, 0, 3, 2, 2, 1, 5, 3, 4, 1, 1, 1
OFFSET
2,6
COMMENTS
Conjecture: every nonnegative integer occurs infinitely many times.
FORMULA
a(n) = A332886(n) - 2 for n >= 2.
EXAMPLE
Let p(n) = A000041(n) = number of partitions of n. Then p(9) = 30, which is divisible by these 6 partition numbers: p(1) = 1, p(2) = 2, p(3) = 3, p(4) = 5, p(7) = 15, and p(9) = 30; discounting p(1) and p(9) leaves a(9) = 4 proper divisors.
MATHEMATICA
p[n_] := PartitionsP[n]; t[n_] := Table[p[k], {k, 0, n}]
u = -2 + Table[Length[Intersection[t[n], Divisors[p[n]]]], {n, 2, 130}]
PROG
(PARI) a(n) = my(nbp=numbpart(n)); sum(k=2, n-1, (nbp % numbpart(k)) == 0); \\ Michel Marcus, Feb 29 2020
CROSSREFS
Cf. A000041 (partition numbers), A332886.
Sequence in context: A105114 A166284 A098086 * A306323 A317015 A341902
KEYWORD
nonn
AUTHOR
Clark Kimberling, Feb 29 2020
STATUS
approved