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

Author

Reinhard Zumkeller, Feb 22 2006

Keywords

• nonn

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.

Examples

			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.
		

Links

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

Crossrefs

Cf. A000005, A051293.

Programs

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 *)

Extensions

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

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