cp's OEIS Frontend

A366845 Number of integer partitions of n that contain at least one even part and whose halved even parts are relatively prime.

%I A366845 #7 Oct 30 2023 11:06:20
%S A366845 0,0,1,1,2,3,5,7,11,15,23,31,43,58,82,107,144,189,250,323,420,537,695,
%T A366845 880,1114,1404,1774,2210,2759,3423,4239,5223,6430,7869,9640,11738,
%U A366845 14266,17297,20950,25256,30423,36545,43824,52421,62620,74599,88802,105431
%N A366845 Number of integer partitions of n that contain at least one even part and whose halved even parts are relatively prime.
%e A366845 The partition y = (6,4) has halved even parts (3,2) which are relatively prime, so y is counted under a(10).
%e A366845 The a(2) = 1 through a(9) = 15 partitions:
%e A366845   (2)  (21)  (22)   (32)    (42)     (52)      (62)       (72)
%e A366845              (211)  (221)   (222)    (322)     (332)      (432)
%e A366845                     (2111)  (321)    (421)     (422)      (522)
%e A366845                             (2211)   (2221)    (521)      (621)
%e A366845                             (21111)  (3211)    (2222)     (3222)
%e A366845                                      (22111)   (3221)     (3321)
%e A366845                                      (211111)  (4211)     (4221)
%e A366845                                                (22211)    (5211)
%e A366845                                                (32111)    (22221)
%e A366845                                                (221111)   (32211)
%e A366845                                                (2111111)  (42111)
%e A366845                                                           (222111)
%e A366845                                                           (321111)
%e A366845                                                           (2211111)
%e A366845                                                           (21111111)
%t A366845 Table[Length[Select[IntegerPartitions[n], GCD@@Select[#,EvenQ]/2==1&]],{n,0,30}]
%Y A366845 For all parts we have A000837, complement A018783.
%Y A366845 These partitions have ranks A366847.
%Y A366845 For odd parts we have A366850, ranks A366846, complement A366842.
%Y A366845 A000041 counts integer partitions, strict A000009, complement A047967.
%Y A366845 A035363 counts partitions into all even parts, ranks A066207.
%Y A366845 A078374 counts relatively prime strict partitions.
%Y A366845 A168532 counts partitions by gcd.
%Y A366845 A239261 counts partitions with (sum of odd parts) = (sum of even parts).
%Y A366845 A366531 = 2*A366533 adds up even prime indices, triangle A113686/A174713.
%Y A366845 Cf. A051424, A113685, A116598, A258117, A289509, A366843, A366849.
%K A366845 nonn
%O A366845 0,5
%A A366845 _Gus Wiseman_, Oct 28 2023