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A357710
Number of integer compositions of n with integer geometric mean.
5
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
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.
Sequence in context: A342330 A358911 A153937 * A242971 A036818 A036813
KEYWORD
nonn
AUTHOR
Gus Wiseman, Oct 15 2022
EXTENSIONS
More terms from David A. Corneth, Oct 17 2022
STATUS
approved