A366750 - OEIS

%I #12 Nov 03 2023 11:21:27

%S 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,

%T 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,

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

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

%e The a(n) partitions for n = 3, 24, 30, 42, 45, 57, 60:

%e   (3)  (15,9)  (21,9)  (33,9)   (45)       (57)       (51,9)

%e        (21,3)  (25,5)  (35,7)   (33,9,3)   (45,9,3)   (55,5)

%e                (27,3)  (39,3)   (21,15,9)  (27,21,9)  (57,3)

%e                        (27,15)  (25,15,5)  (33,15,9)  (33,27)

%e                                 (27,15,3)  (33,21,3)  (35,25)

%e                                            (39,15,3)  (39,21)

%e                                                       (45,15)

%e                                                       (27,21,9,3)

%e                                                       (33,15,9,3)

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

%o (Python)

%o from math import gcd

%o from sympy.utilities.iterables import partitions

%o 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

%Y This is the case of A000700 with a common divisor.

%Y Including evens gives A303280.

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

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

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

%Y A051424 counts pairwise coprime partitions, for odd parts A366853.

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

%Y A168532 counts partitions by gcd.

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

%K nonn

%O 0,25

%A _Gus Wiseman_, Nov 01 2023

%E More terms from _Chai Wah Wu_, Nov 02 2023