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A271654
a(n) = Sum_{k|n} binomial(n-1,k-1).
9
1, 2, 2, 5, 2, 17, 2, 44, 30, 137, 2, 695, 2, 1731, 1094, 6907, 2, 30653, 2, 97244, 38952, 352739, 2, 1632933, 10628, 5200327, 1562602, 20357264, 2, 87716708, 2, 303174298, 64512738, 1166803145, 1391282, 4978661179, 2, 17672631939, 2707475853, 69150651910, 2, 286754260229, 2, 1053966829029, 115133177854, 4116715363847, 2, 16892899722499, 12271514, 63207357886437
OFFSET
1,2
COMMENTS
Also the number of compositions of n whose length divides n, i.e., compositions with integer mean, ranked by A096199. - Gus Wiseman, Sep 28 2022
LINKS
EXAMPLE
From Gus Wiseman, Sep 28 2022: (Start)
The a(1) = 1 through a(6) = 17 compositions with integer mean:
(1) (2) (3) (4) (5) (6)
(1,1) (1,1,1) (1,3) (1,1,1,1,1) (1,5)
(2,2) (2,4)
(3,1) (3,3)
(1,1,1,1) (4,2)
(5,1)
(1,1,4)
(1,2,3)
(1,3,2)
(1,4,1)
(2,1,3)
(2,2,2)
(2,3,1)
(3,1,2)
(3,2,1)
(4,1,1)
(1,1,1,1,1,1)
(End)
MAPLE
a:= n-> add(binomial(n-1, d-1), d=numtheory[divisors](n)):
seq(a(n), n=1..50); # Alois P. Heinz, Dec 03 2023
MATHEMATICA
Table[Length[Join @@ Permutations/@Select[IntegerPartitions[n], IntegerQ[Mean[#]]&]], {n, 15}] (* Gus Wiseman, Sep 28 2022 *)
PROG
(PARI) a(n)=sumdiv(n, k, binomial(n-1, k-1))
CROSSREFS
Cf. A056045.
The version for nonempty subsets is A051293, geometric A326027.
The version for partitions is A067538, ranked by A316413, strict A102627.
These compositions are ranked by A096199.
The version for factorizations is A326622, geometric A326028.
A011782 counts compositions.
A067539 = partitions w integer geo mean, ranked by A326623, strict A326625.
A100346 counts compositions into divisors, partitions A018818.
Sequence in context: A144943 A114976 A085483 * A271622 A324505 A226135
KEYWORD
nonn
AUTHOR
STATUS
approved