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

0,3

Links

Table of n, a(n) for n=0..44.

Example

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)

Mathematica

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

Prog

(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

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.

Sequence in context: A342330 A358911 A153937 * A242971 A036818 A036813

Adjacent sequences: A357707 A357708 A357709 * A357711 A357712 A357713

Keyword

nonn

Author

Gus Wiseman, Oct 15 2022

Extensions

More terms from David A. Corneth, Oct 17 2022

Status

approved