A237984 Number of partitions of n whose mean is a part.

1, 2, 2, 3, 2, 5, 2, 6, 5, 8, 2, 21, 2, 14, 22, 30, 2, 61, 2, 86, 67, 45, 2, 283, 66, 80, 197, 340, 2, 766, 2, 663, 543, 234, 703, 2532, 2, 388, 1395, 4029, 2, 4688, 2, 4476, 7032, 1005, 2, 17883, 2434, 9713, 7684, 14472, 2, 25348, 17562, 37829, 16786, 3721

Offset

1,2

Comments

a(n) = 2 if and only if n is a prime.

Links

Table of n, a(n) for n=1..58.

Formula

a(n) = A000041(n) - A327472(n). - Gus Wiseman, Sep 14 2019

Example

a(6) counts these partitions:  6, 33, 321, 222, 111111.
From Gus Wiseman, Sep 14 2019: (Start)
The a(1) = 1 through a(10) = 8 partitions (A = 10):
  1  2   3    4     5      6       7        8         9          A
     11  111  22    11111  33      1111111  44        333        55
              1111         222              2222      432        22222
                           321              3221      531        32221
                           111111           4211      111111111  33211
                                            11111111             42211
                                                                 52111
                                                                 1111111111
(End)

Mathematica

Table[Count[IntegerPartitions[n], p_ /; MemberQ[p, Mean[p]]], {n, 40}]

Prog

(Python)
from sympy.utilities.iterables import partitions
def A237984(n): return sum(1 for s, p in partitions(n, size=True) if not n%s and n//s in p) # Chai Wah Wu, Sep 21 2023

Crossrefs

Cf. A238478.

The Heinz numbers of these partitions are A327473.

A similar sequence for subsets is A065795.

Dominated by A067538.

The strict case is A240850.

Partitions without their mean are A327472.

Cf. A000016, A316413, A324753, A325705, A327478, A327482.

Sequence in context: A164896 A298422 A304716 * A118136 A356552 A258567

Adjacent sequences: A237981 A237982 A237983 * A237985 A237986 A237987

Keyword

nonn,easy

Author

Clark Kimberling, Feb 27 2014

Status

approved