A366844 Number of strict integer partitions of n into odd relatively prime parts.

0, 1, 0, 0, 1, 0, 1, 0, 2, 1, 2, 1, 2, 2, 3, 3, 5, 4, 4, 5, 6, 7, 8, 8, 9, 11, 12, 12, 15, 16, 15, 19, 23, 23, 26, 28, 30, 34, 37, 38, 44, 48, 48, 56, 62, 63, 72, 77, 82, 92, 96, 102, 116, 124, 128, 142, 155, 162, 178, 191, 200, 222, 236, 246, 276, 291, 303, 334

Offset: 0

Author

Gus Wiseman, Oct 29 2023

Keywords

• nonn

Examples

			The a(n) partitions for n = 1, 8, 14, 17, 16, 20, 21:
  (1)  (5,3)  (9,5)   (9,5,3)   (9,7)      (11,9)      (9,7,5)
       (7,1)  (11,3)  (9,7,1)   (11,5)     (13,7)      (11,7,3)
              (13,1)  (11,5,1)  (13,3)     (17,3)      (11,9,1)
                      (13,3,1)  (15,1)     (19,1)      (13,5,3)
                                (7,5,3,1)  (9,7,3,1)   (13,7,1)
                                           (11,5,3,1)  (15,5,1)
                                                       (17,3,1)
		

Crossrefs

This is the relatively prime case of A000700.

The pairwise coprime version is the odd-part case of A007360.

Allowing even parts gives A078374.

The halved even version is A078374 aerated.

The non-strict version is A366843, with evens A000837.

The complement is counted by the strict case of 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.

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

Cf. A007359, A047967, A055922, A066208, A116598, A239261, A302697, A337485, A365067, 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 A366844(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, Oct 30 2023

Extensions

More terms from Chai Wah Wu, Oct 30 2023