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Number of factorizations of 2^n into factors > 1 with even integer average.
5

%I #5 Jul 18 2019 06:19:03

%S 1,2,2,3,4,5,7,8,11,14,14,20,27,31,41,47,57,75,95,102,155,170,195,239,

%T 327,331,483,517,617,740,952,942,1406,1484,1742,2023,2652,2688,3680,

%U 3892,4729,5375,6689,6911,9437,9938,11754,13529,16710,17419,22346,24230

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

%C 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.

%e The a(1) = 1 through a(8) = 8 partitions:

%e (1) (2) (3) (4) (5) (6) (7) (8)

%e (11) (111) (22) (32) (33) (43) (44)

%e (1111) (311) (42) (52) (53)

%e (11111) (222) (331) (62)

%e (111111) (511) (422)

%e (3211) (2222)

%e (1111111) (4211)

%e (11111111)

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

%Y The strict case is A326670.

%Y Factorizations with integer average are A326622.

%Y Partitions with integer average are A067538.

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

%K nonn

%O 1,2

%A _Gus Wiseman_, Jul 17 2019