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Number of sets of distinct positive integers whose arithmetic mean is an integer, the largest integer of the set being n.
23

%I #42 Feb 22 2023 12:25:03

%S 1,1,3,3,7,11,19,31,59,103,187,343,631,1171,2191,4095,7711,14571,

%T 27595,52431,99879,190651,364723,699071,1342183,2581111,4971067,

%U 9586983,18512791,35791471,69273667,134217727,260301175,505290271,981706831,1908874583,3714566311

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

%C Equivalently, number of nonempty subsets of [n] the sum of whose elements is divisible by n. - _Dimitri Papadopoulos_, Jan 18 2016

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

%F a(n) = A051293(n+1) - A051293(n). - _Reinhard Zumkeller_, Feb 19 2006

%F a(n) = A008965(n) for odd n. - _Dimitri Papadopoulos_, Jan 18 2016

%F 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

%F a(n) = A309402(n,n). - _Alois P. Heinz_, Jul 28 2019

%e 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.

%t Table[Length[Select[Select[Subsets[Range[n]],Max[#]==n&], IntegerQ[ Mean[ #]]&]], {n,22}] (* _Harvey P. Dale_, Jul 23 2011 *)

%t Table[Total[Table[Length[Select[Select[Subsets[Range[n]], Length[#] == k &],IntegerQ[Total[#]/n] &]], {k, n}]], {n, 10}] (* _Dimitri Papadopoulos_, Jan 18 2016 *)

%o (PARI) a(n) = sumdiv(n, d, (d%2)* 2^(n/d)*eulerphi(d))/n - 1; \\ _Michel Marcus_, Feb 10 2016

%o (Python)

%o from sympy import totient, divisors

%o 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

%Y Cf. A008965, A051293, A063776, A309402.

%Y Row sums of A267632.

%K easy,nonn

%O 1,3

%A _Naohiro Nomoto_, May 03 2003

%E a(22) from _Harvey P. Dale_, Jul 23 2011

%E a(23)-a(32) from _Dimitri Papadopoulos_, Jan 18 2016