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

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, 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, 1, 1, 82, 0, 20, 92, 1, 2, 105, 31, 1, 122, 1, 1, 166, 2, 16, 168

Offset: 0

Author

Gus Wiseman, Nov 01 2023

Keywords

• nonn

Examples

			The a(n) partitions for n = 3, 9, 15, 21, 25, 27:
(3)  (9)      (15)         (21)             (25)         (27)
     (3,3,3)  (5,5,5)      (7,7,7)          (15,5,5)     (9,9,9)
              (9,3,3)      (9,9,3)          (5,5,5,5,5)  (15,9,3)
              (3,3,3,3,3)  (15,3,3)                      (21,3,3)
                           (9,3,3,3,3)                   (9,9,3,3,3)
                           (3,3,3,3,3,3,3)               (15,3,3,3,3)
                                                         (9,3,3,3,3,3,3)
                                                         (3,3,3,3,3,3,3,3,3)
		

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..1000

Crossrefs

Allowing even parts gives A018783, complement A000837.

For parts > 1 instead of gcd > 1 we have A087897.

For gcd = 1 instead of gcd > 1 we have A366843.

The strict case is A366750, with evens A303280.

The strict complement is A366844, with evens A078374.

A000041 counts integer partitions, strict A000009 (also into odd parts).

A000700 counts strict partitions into odd parts.

A113685 counts partitions by sum of odd parts, rank statistic A366528.

A168532 counts partitions by gcd.

A366842 counts partitions whose odd parts have a common divisor > 1.

Cf. A007359, A047967, A051424, A055922, A066208, A302698, A337485, A365067, A366845, A366848.

Programs

Mathematica

Table[Length[Select[IntegerPartitions[n],And@@OddQ/@#&&GCD@@#>1&]],{n,15}]

Python

from math import gcd
from sympy.utilities.iterables import partitions
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

Extensions

More terms from Chai Wah Wu, Nov 02 2023

a(0)=0 prepended by Alois P. Heinz, Jan 11 2024