A326671 Number of factorizations of 2^n into factors > 1 with even integer average.

1, 2, 2, 3, 4, 5, 7, 8, 11, 14, 14, 20, 27, 31, 41, 47, 57, 75, 95, 102, 155, 170, 195, 239, 327, 331, 483, 517, 617, 740, 952, 942, 1406, 1484, 1742, 2023, 2652, 2688, 3680, 3892, 4729, 5375, 6689, 6911, 9437, 9938, 11754, 13529, 16710, 17419, 22346, 24230

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 - 1): s in y} is an integer.

Examples

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

Crossrefs

The strict case is A326670.

Factorizations with integer average are A326622.

Partitions with integer average are A067538.

Cf. A001055, A051293, A102627, A326028, A326622, A326647, A326666, A326667, A326668.

Programs

Mathematica

Table[Length[Select[IntegerPartitions[n],IntegerQ[Mean[2^(#-1)]]&]],{n,30}]