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A067538 Number of partitions of n in which the number of parts divides n. 40
1, 2, 2, 4, 2, 8, 2, 11, 9, 14, 2, 46, 2, 24, 51, 66, 2, 126, 2, 202, 144, 69, 2, 632, 194, 116, 381, 756, 2, 1707, 2, 1417, 956, 316, 2043, 5295, 2, 511, 2293, 9151, 2, 10278, 2, 8409, 14671, 1280, 2, 36901, 8035, 21524, 11614, 25639, 2, 53138, 39810, 85004 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,2

COMMENTS

Also sum of p(n,d) over the divisors d of n, where p(n,m) is the count of partitions of n in exactly m parts. - Wouter Meeussen, Jun 07 2009

LINKS

Alois P. Heinz, Table of n, a(n) for n = 1..1000 (first 500 terms from Wouter Meeussen)

Eric W. Weisstein, Partition Function P

Wikipedia, Integer Partition

FORMULA

a(p) = 2 for all primes p.

EXAMPLE

a(3)=2 because 3 is a prime; a(4)=4 because the five partitions of 4 are {4}, {3, 1}, {2, 2}, {2, 1, 1}, {1, 1, 1, 1}, and the number of parts in each of them divides 4 except for {2, 1, 1}.

MATHEMATICA

Do[p = IntegerPartitions[n]; l = Length[p]; c = 0; k = 1; While[k < l + 1, If[ IntegerQ[ n/Length[ p[[k]] ]], c++ ]; k++ ]; Print[c], {n, 1, 57}, All]

p[n_, k_]:=p[n, k]=p[n-1, k-1]+p[n-k, k]; p[n_, k_]:=0/; k>n; p[n_, n_]:=1; p[n_, 0]:=0

Table[Plus @@ (p[n, # ]&/ @ Divisors[n]), {n, 36}] (* Wouter Meeussen, Jun 07 2009 *)

Table[Count[IntegerPartitions[n], q_ /; IntegerQ[Mean[q]]], {n, 50}]  (*Clark Kimberling, Apr 23 2019 *)

PROG

(PARI) a(n) = {my(nb = 0); forpart(p=n, if ((vecsum(Vec(p)) % #p) == 0, nb++); ); nb; } \\ Michel Marcus, Jul 03 2018

CROSSREFS

Cf. A000005, A000041, A143773, A298422, A298423, A298426.

Sequence in context: A100577 A018818 A157019 * A305982 A304102 A096154

Adjacent sequences:  A067535 A067536 A067537 * A067539 A067540 A067541

KEYWORD

easy,nonn

AUTHOR

Naohiro Nomoto, Jan 27 2002

EXTENSIONS

Extended by Robert G. Wilson v, Oct 16 2002

STATUS

approved

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Last modified July 23 11:44 EDT 2019. Contains 325254 sequences. (Running on oeis4.)