A326668 Number of strict factorizations of 2^n into factors > 1 with integer average.

1, 1, 2, 2, 3, 3, 4, 4, 6, 5, 7, 7, 9, 9, 12, 12, 17, 17, 21, 24, 33, 33, 42, 46, 63, 61, 81, 82, 118, 106, 149, 137, 213, 172, 263, 221, 363, 266, 453, 335, 594, 409, 735, 484, 968, 594, 1139, 731, 1486, 813, 1801, 1026, 2177, 1230, 2667, 1348, 3334, 1693

Offset: 1

Author

Gus Wiseman, Jul 17 2019

Keywords

• nonn

Comments

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

Examples

			The a(1) = 1 through a(11) = 7 partitions (A = 10, B = 11):
  (1)  (2)  (3)   (4)   (5)   (6)   (7)   (8)   (9)    (A)   (B)
            (21)  (31)  (32)  (42)  (43)  (53)  (54)   (64)  (65)
                        (41)  (51)  (52)  (62)  (63)   (73)  (74)
                                    (61)  (71)  (72)   (82)  (83)
                                                (81)   (91)  (92)
                                                (531)        (A1)
                                                             (731)
		

Crossrefs

The non-strict case is A326667.

Factorizations with integer average are A326622.

Strict partitions with integer average are A102627.

Subsets with integer average are A051293.

Cf. A001055, A067538, A102627, A326028, A326647, A326666, A326670, A326671.

Programs

Mathematica

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