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%I #20 Jan 11 2024 16:56:50
%S 0,0,0,1,0,1,1,1,0,2,1,1,2,1,1,4,0,1,4,1,2,6,1,1,6,3,1,8,2,1,13,1,0,
%T 13,1,7,15,1,1,19,6,1,25,1,2,33,1,1,32,5,10,39,2,1,46,14,6,55,1,1,77,
%U 1,1,82,0,20,92,1,2,105,31,1,122,1,1,166,2,16,168
%N Number of integer partitions of n into odd parts with a common divisor > 1.
%H Alois P. Heinz, <a href="/A366852/b366852.txt">Table of n, a(n) for n = 0..1000</a>
%e The a(n) partitions for n = 3, 9, 15, 21, 25, 27:
%e (3) (9) (15) (21) (25) (27)
%e (3,3,3) (5,5,5) (7,7,7) (15,5,5) (9,9,9)
%e (9,3,3) (9,9,3) (5,5,5,5,5) (15,9,3)
%e (3,3,3,3,3) (15,3,3) (21,3,3)
%e (9,3,3,3,3) (9,9,3,3,3)
%e (3,3,3,3,3,3,3) (15,3,3,3,3)
%e (9,3,3,3,3,3,3)
%e (3,3,3,3,3,3,3,3,3)
%t Table[Length[Select[IntegerPartitions[n],And@@OddQ/@#&&GCD@@#>1&]],{n,15}]
%o (Python)
%o from math import gcd
%o from sympy.utilities.iterables import partitions
%o def A366852(n): return sum(1 for p in partitions(n) if all(d&1 for d in p) and gcd(*p)>1) # _Chai Wah Wu_, Nov 02 2023
%Y Allowing even parts gives A018783, complement A000837.
%Y For parts > 1 instead of gcd > 1 we have A087897.
%Y For gcd = 1 instead of gcd > 1 we have A366843.
%Y The strict case is A366750, with evens A303280.
%Y The strict complement is A366844, with evens A078374.
%Y A000041 counts integer partitions, strict A000009 (also into odd parts).
%Y A000700 counts strict partitions into odd parts.
%Y A113685 counts partitions by sum of odd parts, rank statistic A366528.
%Y A168532 counts partitions by gcd.
%Y A366842 counts partitions whose odd parts have a common divisor > 1.
%Y Cf. A007359, A047967, A051424, A055922, A066208, A302698, A337485, A365067, A366845, A366848.
%K nonn
%O 0,10
%A _Gus Wiseman_, Nov 01 2023
%E More terms from _Chai Wah Wu_, Nov 02 2023
%E a(0)=0 prepended by _Alois P. Heinz_, Jan 11 2024