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

Author

Clark Kimberling, Feb 27 2014

Keywords

• nonn
• easy

Comments

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

Examples

			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)
		

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.

Programs

Mathematica

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

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

Formula

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