A114976 Number of subsets of {1,2,....,n} with an arithmetic mean that is an integer and also a divisor of n.

1, 2, 2, 5, 2, 14, 2, 30, 11, 80, 2, 280, 2, 764, 128, 2557, 2, 9036, 2, 29656, 1958, 103134, 2, 373454, 119, 1300824, 36992, 4681568, 2, 17119030, 2, 61799636, 758982, 226451040, 2180, 837469677, 2, 3084255132, 16391220, 11451833394, 2, 42746493556, 2

Offset

1,2

Comments

a(n) <= A051293(n);

a(n) = 2 iff n is prime, just as for the number of divisors of n and also, at least for the very first terms, a(n)=odd iff n is a square: these observations might suggest conjectures on a deeper relationship with A000005.

Links

Alois P. Heinz, Table of n, a(n) for n = 1..100

Example

a(9) = 11: {1}, {3}, {9}, {1,5}, {2,4}, {1,2,6}, {1,3,5}, {2,3,4}, {1,2,3,6}, {1,2,4,5} and {1,2,3,4,5}, e.g. also {1,4,7} has an integral arithmetic mean, but (1+4+7)/3 = 4 is not a divisor of 9.

Maple

b:= proc(n, m, s, c) option remember; `if`(n=0,
      `if`(c>0 and denom(s)=1 and irem(m, s)=0, 1, 0),
       b(n-1, m, s, c)+b(n-1, m, (s*c+n)/(c+1), c+1))
    end:
a:= proc(n) option remember; forget (b); b(n$2, 0$2) end:
seq(a(n), n=1..50);  # Alois P. Heinz, Jul 15 2019

Mathematica

b[n_, m_, s_, c_] := b[n, m, s, c] = If[n==0, If[c>0 && Denominator[s]==1 && Mod[m, s]==0, 1, 0], b[n-1, m, s, c]+b[n-1, m, (s c + n)/(c+1), c+1]];
a[n_] := b[n, n, 0, 0];
Array[a, 50] (* Jean-François Alcover, Nov 04 2020, after Alois P. Heinz *)

Crossrefs

Cf. A000005, A051293.

Sequence in context: A068058 A192233 A144943 * A085483 A271654 A271622

Adjacent sequences: A114973 A114974 A114975 * A114977 A114978 A114979

Keyword

nonn

Author

Reinhard Zumkeller, Feb 22 2006

Extensions

a(27)-a(38) from Donovan Johnson, Jun 10 2010

a(39)-a(43) from Alois P. Heinz, Jul 15 2019

Status

approved