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

0,5

Links

Table of n, a(n) for n=0..55.

Example

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)

Mathematica

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

Prog

(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

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.

Sequence in context: A093594 A008806 A356607 * A370805 A238860 A144367

Adjacent sequences: A366840 A366841 A366842 * A366844 A366845 A366846

Keyword

nonn

Author

Gus Wiseman, Oct 28 2023

Status

approved