A349156 Number of integer partitions of n whose mean is not an integer.

1, 0, 0, 1, 1, 5, 3, 13, 11, 21, 28, 54, 31, 99, 111, 125, 165, 295, 259, 488, 425, 648, 933, 1253, 943, 1764, 2320, 2629, 2962, 4563, 3897, 6840, 6932, 9187, 11994, 12840, 12682, 21635, 25504, 28892, 28187, 44581, 42896, 63259, 66766, 74463, 104278, 124752

Offset: 0

Author

Gus Wiseman, Nov 14 2021

Keywords

• nonn

Comments

Equivalently, partitions whose length does not divide their sum.

By conjugation, also the number of integer partitions of n with greatest part not dividing n.

Examples

			The a(3) = 1 through a(8) = 11 partitions:
  (21)  (211)  (32)    (2211)   (43)      (332)
               (41)    (3111)   (52)      (422)
               (221)   (21111)  (61)      (431)
               (311)            (322)     (521)
               (2111)           (331)     (611)
                                (421)     (22211)
                                (511)     (32111)
                                (2221)    (41111)
                                (3211)    (221111)
                                (4111)    (311111)
                                (22111)   (2111111)
                                (31111)
                                (211111)
		

Crossrefs

Below, "!" means either enumerative or set theoretical complement.

The version for nonempty subsets is !A051293.

The complement is counted by A067538, ranked by A316413.

The geometric version is !A067539, strict !A326625, ranked by !A326623.

The strict case is !A102627.

The version for prime factors is A175352, complement A078175.

The version for distinct prime factors is A176587, complement A078174.

The ordered version (compositions) is !A271654, ranked by !A096199.

The multiplicative version (factorizations) is !A326622, geometric !A326028.

The conjugate is ranked by !A326836.

The conjugate strict version is !A326850.

These partitions are ranked by A348551.

A000041 counts integer partitions.

A326567/A326568 give the mean of prime indices, conjugate A326839/A326840.

A236634 counts unbalanced partitions, complement of A047993.

A327472 counts partitions not containing their mean, complement of A237984.

Cf. A001700, A074761, A098743, A143773, A175397, A175761, A298423, A326027, A326641, A326842, A326849, A327778.

Programs

Mathematica

Table[Length[Select[IntegerPartitions[n],!IntegerQ[Mean[#]]&]],{n,0,30}]

Formula

a(n > 0) = A000041(n) - A067538(n).