A067538 Number of partitions of n in which the number of parts divides n.

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

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

From Gus Wiseman, Sep 24 2019: (Start)

Also the number of integer partitions of n whose maximum part divides n. The Heinz numbers of these partitions are given by A326836. For example, the a(1) = 1 through a(8) = 11 partitions are:

(1) (2) (3) (4) (5) (6) (7) (8)

(11) (111) (22) (11111) (33) (1111111) (44)

(211) (222) (422)

(1111) (321) (431)

(2211) (2222)

(3111) (4211)

(21111) (22211)

(111111) (41111)

(221111)

(2111111)

(11111111)

(End)

Links

Chai Wah Wu, Table of n, a(n) for n = 1..10000 (n = 1..500 from Wouter Meeussen, n = 501..1000 from Alois P. Heinz, n = 1001..5000 from David A. Corneth)

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}.
From Gus Wiseman, Sep 24 2019: (Start)
The a(1) = 1 through a(8) = 11 partitions whose length divides their sum are the following. The Heinz numbers of these partitions are given by A316413.
  (1)  (2)   (3)    (4)     (5)      (6)       (7)        (8)
       (11)  (111)  (22)    (11111)  (33)      (1111111)  (44)
                    (31)             (42)                 (53)
                    (1111)           (51)                 (62)
                                     (222)                (71)
                                     (321)                (2222)
                                     (411)                (3221)
                                     (111111)             (3311)
                                                          (4211)
                                                          (5111)
                                                          (11111111)
(End)

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
(Python)
# uses A008284_T
from sympy import divisors
def A067538(n): return sum(A008284_T(n, d) for d in divisors(n, generator=True)) # Chai Wah Wu, Sep 21 2023

Crossrefs

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

The strict case is A102627.

Partitions with integer geometric mean are A067539.

Cf. A018818, A102627, A316413, A326622, A326836, A326843, A326850.

Sequence in context: A018818 A157019 A359906 * 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