A237984 - OEIS

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

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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