cp's OEIS Frontend

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

%I A366842 #10 Oct 28 2023 23:53:01
%S A366842 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,
%T A366842 140,359,201,496,299,685,410,949,575,1284,804,1726,1093,2327,1482,
%U A366842 3076,2023,4060,2684,5358,3572,6970,4745,9050,6221,11734,8115,15060,10609
%N A366842 Number of integer partitions of n whose odd parts have a common divisor > 1.
%e A366842 The a(3) = 1 through a(11) = 13 partitions:
%e A366842   (3)  .  (5)    (3,3)  (7)      (3,3,2)  (9)        (5,5)      (11)
%e A366842           (3,2)         (4,3)             (5,4)      (4,3,3)    (6,5)
%e A366842                         (5,2)             (6,3)      (3,3,2,2)  (7,4)
%e A366842                         (3,2,2)           (7,2)                 (8,3)
%e A366842                                           (3,3,3)               (9,2)
%e A366842                                           (4,3,2)               (4,4,3)
%e A366842                                           (5,2,2)               (5,4,2)
%e A366842                                           (3,2,2,2)             (6,3,2)
%e A366842                                                                 (7,2,2)
%e A366842                                                                 (3,3,3,2)
%e A366842                                                                 (4,3,2,2)
%e A366842                                                                 (5,2,2,2)
%e A366842                                                                 (3,2,2,2,2)
%t A366842 Table[Length[Select[IntegerPartitions[n], GCD@@Select[#,OddQ]>1&]], {n,0,30}]
%o A366842 (Python)
%o A366842 from math import gcd
%o A366842 from sympy.utilities.iterables import partitions
%o A366842 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
%Y A366842 This is the odd case of A018783, complement A000837.
%Y A366842 The even version is A047967.
%Y A366842 The complement is counted by A366850, ranks A366846.
%Y A366842 A000041 counts integer partitions, strict A000009.
%Y A366842 A000740 counts relatively prime compositions.
%Y A366842 A113685 counts partitions by sum of odds, stat A366528, w/o zeros A365067.
%Y A366842 A168532 counts partitions by gcd.
%Y A366842 A239261 counts partitions with (sum of odd parts) = (sum of even parts).
%Y A366842 A289508 gives gcd of prime indices, positions of ones A289509.
%Y A366842 Cf. A007359, A051424, A055922, A066208, A078374, A087436, A116598, A337485, A366843, A366844, A366845.
%K A366842 nonn
%O A366842 0,6
%A A366842 _Gus Wiseman_, Oct 28 2023