A271654 a(n) = Sum_{k|n} binomial(n-1,k-1).

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

Alois P. Heinz, Table of n, a(n) for n = 1..3329

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.

Cf. A000041, A078174, A078175, A326567/A326568, A326624, A326641, A326836, A326837, A326843.

Sequence in context: A144943 A114976 A085483 * A271622 A324505 A226135

Adjacent sequences: A271651 A271652 A271653 * A271655 A271656 A271657

Keyword

nonn

Author

Franklin T. Adams-Watters, Apr 11 2016

Status

approved