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A366852 Number of integer partitions of n into odd parts with a common divisor > 1.

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