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

Author

Franklin T. Adams-Watters, Apr 11 2016

Keywords

• nonn

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

Examples

			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)
		

Links

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

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.

Programs

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 *)

PARI

a(n)=sumdiv(n,k,binomial(n-1,k-1))