A366852 - OEIS

A366852

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

%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