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

1,3

Comments

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

Links

Table of n, a(n) for n=1..37.

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

Example

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.

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 *)

Prog

(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//d-1 for d in divisors(n>>(~n&n-1).bit_length(), generator=True))<<1)//n-1 # Chai Wah Wu, Feb 22 2023

Crossrefs

Cf. A008965, A051293, A063776, A309402.

Row sums of A267632.

Sequence in context: A362055 A358827 A022403 * A327397 A036056 A339586

Adjacent sequences: A082547 A082548 A082549 * A082551 A082552 A082553

Keyword

easy,nonn

Author

Naohiro Nomoto, May 03 2003

Extensions

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

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

Status

approved