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The number of partitions of n in which at least one part is a multiple of 3.
4

%I #16 May 24 2022 08:13:27

%S 0,0,0,1,1,2,4,6,9,14,20,29,41,57,78,106,142,189,250,327,425,549,705,

%T 900,1144,1445,1819,2279,2844,3534,4379,5403,6648,8152,9969,12152,

%U 14780,17920,21682,26163,31504,37842,45371,54270,64800,77211,91842,109031,129235,152897

%N The number of partitions of n in which at least one part is a multiple of 3.

%C From _Gus Wiseman_, May 23 2022: (Start)

%C Also the number of integer partitions of n with at least one part appearing more than twice. The Heinz numbers of these partitions are given by A046099. For example, the a(0) = 0 though a(8) = 9 partitions are:

%C . . . (111) (1111) (2111) (222) (2221) (2222)

%C (11111) (3111) (4111) (5111)

%C (21111) (22111) (22211)

%C (111111) (31111) (32111)

%C (211111) (41111)

%C (1111111) (221111)

%C (311111)

%C (2111111)

%C (11111111)

%C (End)

%F a(n) = A000041(n)-A000726(n).

%e From _Gus Wiseman_, May 23 2022: (Start)

%e The a(0) = 0 through a(8) = 9 partitions with a part that is a multiple of 3:

%e . . . (3) (31) (32) (6) (43) (53)

%e (311) (33) (61) (62)

%e (321) (322) (332)

%e (3111) (331) (431)

%e (3211) (611)

%e (31111) (3221)

%e (3311)

%e (32111)

%e (311111)

%e (End)

%t Table[Length[Select[IntegerPartitions[n],MemberQ[#/3,_?IntegerQ]&]],{n,0,30}] (* _Gus Wiseman_, May 23 2022 *)

%t Table[Length[Select[IntegerPartitions[n],MatchQ[#,{___,x_,x_,x_,___}]&]],{n,0,30}] (* _Gus Wiseman_, May 23 2022 *)

%Y The complement is counted by A000726, ranked by A004709.

%Y These partitions are ranked by A354235.

%Y This is column k = 3 of A354234.

%Y For 2 instead of 3 we have A047967, ranked by A013929 and A324929.

%Y For 4 instead of 3 we have A295342, ranked by A046101.

%Y A000041 counts integer partitions, strict A000009.

%Y A046099 lists non-cubefree numbers.

%Y Cf. A001522, A006918, A064410, A064428, A117485, A188674, A325187, A325534.

%K nonn

%O 0,6

%A _R. J. Mathar_, Nov 20 2017