A067539 Number of partitions of n in which, if the number of parts is k, the product of the parts is the k-th power of some positive integer.

1, 2, 2, 3, 3, 4, 3, 4, 4, 8, 3, 8, 5, 7, 8, 8, 7, 9, 8, 17, 11, 11, 8, 16, 17, 17, 14, 18, 17, 26, 19, 24, 20, 30, 28, 32, 27, 37, 35, 48, 37, 45, 37, 51, 51, 58, 50, 64, 62, 83, 73, 84, 69, 91, 89, 101, 97, 116, 111, 136, 123, 142, 138, 160, 161, 181, 171, 205, 199, 231, 221

Offset: 1

Author

Naohiro Nomoto, Jan 27 2002

Keywords

• easy
• nonn

Comments

a(n) is the number of integer partitions of n whose geometric mean is an integer. - Gus Wiseman, Jul 19 2019

Examples

			From _Gus Wiseman_, Jul 19 2019: (Start)
The a(1) = 1 through a(8) = 4 partitions:
  (1)  (2)   (3)    (4)     (5)      (6)       (7)        (8)
       (11)  (111)  (22)    (41)     (33)      (421)      (44)
                    (1111)  (11111)  (222)     (1111111)  (2222)
                                     (111111)             (11111111)
(End)
		

Links

• Wikipedia, Geometric mean

Crossrefs

Partitions with integer average are A067538.

Subsets whose geometric mean is an integer are A326027.

The Heinz numbers of these partitions are A326623.

The strict case is A326625.

Cf. A000041, A102627, A320322, A326028, A326641.

Programs

Mathematica

Table[Length[Select[IntegerPartitions[n],IntegerQ[GeometricMean[#]]&]],{n,30}] (* Gus Wiseman, Jul 19 2019 *)

Python

from math import prod
from sympy import integer_nthroot
from sympy.utilities.iterables import partitions
def A067539(n): return sum(1 for s,p in partitions(n,size=True) if integer_nthroot(prod(a**b for a, b in p.items()),s)[1]) # Chai Wah Wu, Sep 24 2023

Extensions

Terms a(61) onwards from Max Alekseyev, Feb 06 2010