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%I #12 Nov 03 2023 11:21:27
%S 0,0,0,1,0,1,0,1,0,1,0,1,1,1,0,1,0,1,1,1,1,1,0,1,2,1,0,2,1,1,3,1,0,2,
%T 0,1,3,1,0,3,2,1,4,1,1,5,0,1,5,1,2,5,1,1,5,2,2,6,0,1,9,1,0,9,0,3,9,1,
%U 1,9,5,1,11,1,0,15,1,2,13,1,5,14,0,1,18
%N Number of strict integer partitions of n into odd parts with a common divisor > 1.
%e The a(n) partitions for n = 3, 24, 30, 42, 45, 57, 60:
%e (3) (15,9) (21,9) (33,9) (45) (57) (51,9)
%e (21,3) (25,5) (35,7) (33,9,3) (45,9,3) (55,5)
%e (27,3) (39,3) (21,15,9) (27,21,9) (57,3)
%e (27,15) (25,15,5) (33,15,9) (33,27)
%e (27,15,3) (33,21,3) (35,25)
%e (39,15,3) (39,21)
%e (45,15)
%e (27,21,9,3)
%e (33,15,9,3)
%t Table[Length[Select[IntegerPartitions[n], And@@OddQ/@#&&UnsameQ@@#&&GCD@@#>1&]], {n,0,30}]
%o (Python)
%o from math import gcd
%o from sympy.utilities.iterables import partitions
%o def A366750(n): return sum(1 for p in partitions(n) if all(d==1 for d in p.values()) and all(d&1 for d in p) and gcd(*p)>1) # _Chai Wah Wu_, Nov 02 2023
%Y This is the case of A000700 with a common divisor.
%Y Including evens gives A303280.
%Y The complement is counted by A366844, non-strict version A366843.
%Y The non-strict version is A366852, with evens A018783.
%Y A000041 counts integer partitions, strict A000009 (also into odds).
%Y A051424 counts pairwise coprime partitions, for odd parts A366853.
%Y A113685 counts partitions by sum of odd parts, rank statistic A366528.
%Y A168532 counts partitions by gcd.
%Y Cf. A000837, A007360, A055922, A066208, A078374, A302697, A337485, A366842, A366845, A366848.
%K nonn
%O 0,25
%A _Gus Wiseman_, Nov 01 2023
%E More terms from _Chai Wah Wu_, Nov 02 2023
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