A082550 Number of sets of distinct positive integers whose arithmetic mean is an integer, the largest integer of the set being n.

1, 1, 3, 3, 7, 11, 19, 31, 59, 103, 187, 343, 631, 1171, 2191, 4095, 7711, 14571, 27595, 52431, 99879, 190651, 364723, 699071, 1342183, 2581111, 4971067, 9586983, 18512791, 35791471, 69273667, 134217727, 260301175, 505290271, 981706831, 1908874583, 3714566311

Offset: 1

Author

Naohiro Nomoto, May 03 2003

Keywords

• easy
• nonn

Comments

Equivalently, number of nonempty subsets of [n] the sum of whose elements is divisible by n. - Dimitri Papadopoulos, Jan 18 2016

Examples

			a(5) = 7: the seven sets are (1+2+3+4+5)/5 = 3, 5/1 = 5, (1+5)/2 = 3, (1+3+5)/3 = 3, (3+5)/2 = 4, (3+4+5)/3 = 4, (1+2+4+5)/4 = 3.
		

Crossrefs

Cf. A008965, A051293, A063776, A309402.

Row sums of A267632.

Programs

Mathematica

Table[Length[Select[Select[Subsets[Range[n]],Max[#]==n&], IntegerQ[ Mean[ #]]&]], {n,22}] (* Harvey P. Dale, Jul 23 2011 *)
Table[Total[Table[Length[Select[Select[Subsets[Range[n]], Length[#] == k &],IntegerQ[Total[#]/n] &]], {k, n}]], {n, 10}] (* Dimitri Papadopoulos, Jan 18 2016 *)

PARI

a(n) = sumdiv(n, d, (d%2)* 2^(n/d)*eulerphi(d))/n - 1; \\ Michel Marcus, Feb 10 2016

Python

from sympy import totient, divisors
def A082550(n): return (sum(totient(d)<>(~n&n-1).bit_length(),generator=True))<<1)//n-1 # Chai Wah Wu, Feb 22 2023

Formula

a(n) = A063776(n) - 1.

a(n) = A051293(n+1) - A051293(n). - Reinhard Zumkeller, Feb 19 2006

a(n) = A008965(n) for odd n. - Dimitri Papadopoulos, Jan 18 2016

G.f.: -x/(1 - x) - Sum_{m >= 0} (phi(2*m + 1)/(2*m + 1)) * log(1 - 2*x^(2*m + 1)). - Petros Hadjicostas, Jul 13 2019

a(n) = A309402(n,n). - Alois P. Heinz, Jul 28 2019

Extensions

a(22) from Harvey P. Dale, Jul 23 2011

a(23)-a(32) from Dimitri Papadopoulos, Jan 18 2016