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

1,2

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.

Links

Table of n, a(n) for n=1..52.

Example

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)

Mathematica

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

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.

Sequence in context: A304329 A111901 A316202 * A134727 A152305 A131419

Adjacent sequences: A326668 A326669 A326670 * A326672 A326673 A326674

Keyword

nonn

Author

Gus Wiseman, Jul 17 2019

Status

approved