A295341 The number of partitions of n in which at least one part is a multiple of 3.

0, 0, 0, 1, 1, 2, 4, 6, 9, 14, 20, 29, 41, 57, 78, 106, 142, 189, 250, 327, 425, 549, 705, 900, 1144, 1445, 1819, 2279, 2844, 3534, 4379, 5403, 6648, 8152, 9969, 12152, 14780, 17920, 21682, 26163, 31504, 37842, 45371, 54270, 64800, 77211, 91842, 109031, 129235, 152897

Offset: 0

Author

R. J. Mathar, Nov 20 2017

Keywords

• nonn

Comments

From Gus Wiseman, May 23 2022: (Start)

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:

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

(11111) (3111) (4111) (5111)

(21111) (22111) (22211)

(111111) (31111) (32111)

(211111) (41111)

(1111111) (221111)

(311111)

(2111111)

(11111111)

(End)

Examples

			From _Gus Wiseman_, May 23 2022: (Start)
The a(0) = 0 through a(8) = 9 partitions with a part that is a multiple of 3:
  .  .  .  (3)  (31)  (32)   (6)     (43)     (53)
                      (311)  (33)    (61)     (62)
                             (321)   (322)    (332)
                             (3111)  (331)    (431)
                                     (3211)   (611)
                                     (31111)  (3221)
                                              (3311)
                                              (32111)
                                              (311111)
(End)
		

Crossrefs

The complement is counted by A000726, ranked by A004709.

These partitions are ranked by A354235.

This is column k = 3 of A354234.

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

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

A000041 counts integer partitions, strict A000009.

A046099 lists non-cubefree numbers.

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

Programs

Mathematica

Table[Length[Select[IntegerPartitions[n],MemberQ[#/3,?IntegerQ]&]],{n,0,30}] (* _Gus Wiseman, May 23 2022 *)
Table[Length[Select[IntegerPartitions[n],MatchQ[#,{_,x_,x_,x_,_}]&]],{n,0,30}] (* Gus Wiseman, May 23 2022 *)

Formula

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