A366843 Number of integer partitions of n into odd, relatively prime parts.

1, 1, 1, 1, 2, 2, 3, 4, 6, 6, 9, 11, 13, 17, 21, 23, 32, 37, 42, 53, 62, 70, 88, 103, 116, 139, 164, 184, 220, 255, 283, 339, 390, 435, 511, 578, 653, 759, 863, 963, 1107, 1259, 1401, 1609, 1814, 2015, 2303, 2589, 2878, 3259, 3648, 4058, 4580, 5119, 5672, 6364

Offset: 0

Author

Gus Wiseman, Oct 28 2023

Keywords

• nonn

Examples

			The a(1) = 1 through a(8) = 6 partitions:
  (1)  (11)  (111)  (31)    (311)    (51)      (331)      (53)
                    (1111)  (11111)  (3111)    (511)      (71)
                                     (111111)  (31111)    (3311)
                                               (1111111)  (5111)
                                                          (311111)
                                                          (11111111)
		

Crossrefs

Allowing even parts gives A000837.

The strict case is A366844, with evens A078374.

The complement is counted by A366852, with evens A018783.

The pairwise coprime version is A366853, with evens A051424.

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

A000740 counts relatively prime compositions.

A168532 counts partitions by gcd.

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

Cf. A007359, A047967, A055922, A066208, A113685, A116598, A289509, A289508, A302697, A337485, A366845, A366848, A366849.

Programs

Mathematica

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

Python

from math import gcd
from sympy.utilities.iterables import partitions
def A366843(n): return sum(1 for p in partitions(n) if all(d&1 for d in p) and gcd(*p)==1) # Chai Wah Wu, Oct 30 2023