A326667 Number of factorizations of 2^n into factors > 1 with integer average.

1, 2, 3, 4, 5, 7, 8, 11, 15, 19, 21, 29, 37, 44, 58, 67, 86, 105, 136, 146, 219, 236, 295, 327, 473, 469, 694, 707, 932, 1020, 1398, 1340, 2023, 2059, 2636, 2816, 3887, 3855, 5377, 5467, 7095, 7611, 9924, 9992, 13795, 14205, 17728, 19315, 24803, 25452, 33026

Offset: 1

Author

Gus Wiseman, Jul 17 2019

Keywords

• nonn

Comments

Also the number of integer partitions y of n such that the average of the multiset {2^s: s in y} is an integer.

Examples

			The a(1) = 1 through a(8) = 11 partitions:
  (1)  (2)   (3)    (4)     (5)      (6)       (7)        (8)
       (11)  (21)   (22)    (32)     (33)      (43)       (44)
             (111)  (31)    (41)     (42)      (52)       (53)
                    (1111)  (311)    (51)      (61)       (62)
                            (11111)  (222)     (331)      (71)
                                     (2211)    (511)      (422)
                                     (111111)  (3211)     (2222)
                                               (1111111)  (3311)
                                                          (4211)
                                                          (311111)
                                                          (11111111)
		

Crossrefs

The strict case is A326668.

Factorizations with integer average are A326622.

Partitions with integer average are A067538.

Subsets with integer average are A051293.

Cf. A001055, A102627, A326028, A326647, A326666, A326670, A326671.

Programs

Mathematica

Table[Length[Select[IntegerPartitions[n],IntegerQ[Mean[2^#]]&]],{n,30}]