A366842 Number of integer partitions of n whose odd parts have a common divisor > 1.

0, 0, 0, 1, 0, 2, 1, 4, 1, 8, 3, 13, 6, 21, 10, 36, 15, 53, 28, 80, 41, 122, 63, 174, 97, 250, 140, 359, 201, 496, 299, 685, 410, 949, 575, 1284, 804, 1726, 1093, 2327, 1482, 3076, 2023, 4060, 2684, 5358, 3572, 6970, 4745, 9050, 6221, 11734, 8115, 15060, 10609

Offset

0,6

Links

Table of n, a(n) for n=0..54.

Example

The a(3) = 1 through a(11) = 13 partitions:
  (3)  .  (5)    (3,3)  (7)      (3,3,2)  (9)        (5,5)      (11)
          (3,2)         (4,3)             (5,4)      (4,3,3)    (6,5)
                        (5,2)             (6,3)      (3,3,2,2)  (7,4)
                        (3,2,2)           (7,2)                 (8,3)
                                          (3,3,3)               (9,2)
                                          (4,3,2)               (4,4,3)
                                          (5,2,2)               (5,4,2)
                                          (3,2,2,2)             (6,3,2)
                                                                (7,2,2)
                                                                (3,3,3,2)
                                                                (4,3,2,2)
                                                                (5,2,2,2)
                                                                (3,2,2,2,2)

Mathematica

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

Prog

(Python)
from math import gcd
from sympy.utilities.iterables import partitions
def A366842(n): return sum(1 for p in partitions(n) if gcd(*(q for q in p if q&1))>1) # Chai Wah Wu, Oct 28 2023

Crossrefs

This is the odd case of A018783, complement A000837.

The even version is A047967.

The complement is counted by A366850, ranks A366846.

A000041 counts integer partitions, strict A000009.

A000740 counts relatively prime compositions.

A113685 counts partitions by sum of odds, stat A366528, w/o zeros A365067.

A168532 counts partitions by gcd.

A239261 counts partitions with (sum of odd parts) = (sum of even parts).

A289508 gives gcd of prime indices, positions of ones A289509.

Cf. A007359, A051424, A055922, A066208, A078374, A087436, A116598, A337485, A366843, A366844, A366845.

Sequence in context: A248058 A072345 A200583 * A115120 A147373 A147441

Adjacent sequences: A366839 A366840 A366841 * A366843 A366844 A366845

Keyword

nonn

Author

Gus Wiseman, Oct 28 2023

Status

approved