A357710 Number of integer compositions of n with integer geometric mean.

0, 1, 2, 2, 3, 4, 4, 8, 4, 15, 17, 22, 48, 40, 130, 88, 287, 323, 543, 1084, 1145, 2938, 3141, 6928, 9770, 15585, 29249, 37540, 78464, 103289, 194265, 299752, 475086, 846933, 1216749, 2261920, 3320935, 5795349, 9292376, 14825858, 25570823, 39030115, 68265801, 106030947, 178696496

Offset: 0

Author

Gus Wiseman, Oct 15 2022

Keywords

• nonn

Examples

			The a(6) = 4 through a(9) = 15 compositions:
  (6)       (7)        (8)         (9)
  (33)      (124)      (44)        (333)
  (222)     (142)      (2222)      (1224)
  (111111)  (214)      (11111111)  (1242)
            (241)                  (1422)
            (412)                  (2124)
            (421)                  (2142)
            (1111111)              (2214)
                                   (2241)
                                   (2412)
                                   (2421)
                                   (4122)
                                   (4212)
                                   (4221)
                                   (111111111)
		

Crossrefs

The unordered version (partitions) is A067539, ranked by A326623.

Compositions with integer average are A271654, partitions A067538.

Subsets whose geometric mean is an integer are A326027.

The version for factorizations is A326028.

The strict case is A339452, partitions A326625.

These compositions are ranked by A357490.

A011782 counts compositions.

Cf. A025047, A051293, A078174, A078175, A102627, A320322, A326622, A326624, A326641, A357182, A357183.

Programs

Mathematica

Table[Length[Select[Join @@ Permutations/@IntegerPartitions[n],IntegerQ[GeometricMean[#]]&]],{n,0,15}]

Python

from math import prod, factorial
from sympy import integer_nthroot
from sympy.utilities.iterables import partitions
def A357710(n): return sum(factorial(s)//prod(factorial(d) for d in p.values()) for s,p in partitions(n,size=True) if integer_nthroot(prod(a**b for a, b in p.items()),s)[1]) if n else 0 # Chai Wah Wu, Sep 24 2023

Extensions

More terms from David A. Corneth, Oct 17 2022