cp's OEIS Frontend

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

%I A357710 #15 Sep 24 2023 13:03:43
%S A357710 0,1,2,2,3,4,4,8,4,15,17,22,48,40,130,88,287,323,543,1084,1145,2938,
%T A357710 3141,6928,9770,15585,29249,37540,78464,103289,194265,299752,475086,
%U A357710 846933,1216749,2261920,3320935,5795349,9292376,14825858,25570823,39030115,68265801,106030947,178696496
%N A357710 Number of integer compositions of n with integer geometric mean.
%e A357710 The a(6) = 4 through a(9) = 15 compositions:
%e A357710   (6)       (7)        (8)         (9)
%e A357710   (33)      (124)      (44)        (333)
%e A357710   (222)     (142)      (2222)      (1224)
%e A357710   (111111)  (214)      (11111111)  (1242)
%e A357710             (241)                  (1422)
%e A357710             (412)                  (2124)
%e A357710             (421)                  (2142)
%e A357710             (1111111)              (2214)
%e A357710                                    (2241)
%e A357710                                    (2412)
%e A357710                                    (2421)
%e A357710                                    (4122)
%e A357710                                    (4212)
%e A357710                                    (4221)
%e A357710                                    (111111111)
%t A357710 Table[Length[Select[Join @@ Permutations/@IntegerPartitions[n],IntegerQ[GeometricMean[#]]&]],{n,0,15}]
%o A357710 (Python)
%o A357710 from math import prod, factorial
%o A357710 from sympy import integer_nthroot
%o A357710 from sympy.utilities.iterables import partitions
%o A357710 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
%Y A357710 The unordered version (partitions) is A067539, ranked by A326623.
%Y A357710 Compositions with integer average are A271654, partitions A067538.
%Y A357710 Subsets whose geometric mean is an integer are A326027.
%Y A357710 The version for factorizations is A326028.
%Y A357710 The strict case is A339452, partitions A326625.
%Y A357710 These compositions are ranked by A357490.
%Y A357710 A011782 counts compositions.
%Y A357710 Cf. A025047, A051293, A078174, A078175, A102627, A320322, A326622, A326624, A326641, A357182, A357183.
%K A357710 nonn
%O A357710 0,3
%A A357710 _Gus Wiseman_, Oct 15 2022
%E A357710 More terms from _David A. Corneth_, Oct 17 2022