A366750 Number of strict integer partitions of n into odd parts with a common divisor > 1.

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, 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, 1, 9, 5, 1, 11, 1, 0, 15, 1, 2, 13, 1, 5, 14, 0, 1, 18

Offset: 0

Author

Gus Wiseman, Nov 01 2023

Keywords

• nonn

Examples

			The a(n) partitions for n = 3, 24, 30, 42, 45, 57, 60:
  (3)  (15,9)  (21,9)  (33,9)   (45)       (57)       (51,9)
       (21,3)  (25,5)  (35,7)   (33,9,3)   (45,9,3)   (55,5)
               (27,3)  (39,3)   (21,15,9)  (27,21,9)  (57,3)
                       (27,15)  (25,15,5)  (33,15,9)  (33,27)
                                (27,15,3)  (33,21,3)  (35,25)
                                           (39,15,3)  (39,21)
                                                      (45,15)
                                                      (27,21,9,3)
                                                      (33,15,9,3)
		

Crossrefs

This is the case of A000700 with a common divisor.

Including evens gives A303280.

The complement is counted by A366844, non-strict version A366843.

The non-strict version is A366852, with evens A018783.

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

A051424 counts pairwise coprime partitions, for odd parts A366853.

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

A168532 counts partitions by gcd.

Cf. A000837, A007360, A055922, A066208, A078374, A302697, A337485, A366842, A366845, A366848.

Programs

Mathematica

Table[Length[Select[IntegerPartitions[n], And@@OddQ/@#&&UnsameQ@@#&&GCD@@#>1&]], {n,0,30}]

Python

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

Extensions

More terms from Chai Wah Wu, Nov 02 2023