cp's OEIS Frontend

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

%I A082550 #42 Feb 22 2023 12:25:03
%S A082550 1,1,3,3,7,11,19,31,59,103,187,343,631,1171,2191,4095,7711,14571,
%T A082550 27595,52431,99879,190651,364723,699071,1342183,2581111,4971067,
%U A082550 9586983,18512791,35791471,69273667,134217727,260301175,505290271,981706831,1908874583,3714566311
%N A082550 Number of sets of distinct positive integers whose arithmetic mean is an integer, the largest integer of the set being n.
%C A082550 Equivalently, number of nonempty subsets of [n] the sum of whose elements is divisible by n. - _Dimitri Papadopoulos_, Jan 18 2016
%F A082550 a(n) = A063776(n) - 1.
%F A082550 a(n) = A051293(n+1) - A051293(n). - _Reinhard Zumkeller_, Feb 19 2006
%F A082550 a(n) = A008965(n) for odd n. - _Dimitri Papadopoulos_, Jan 18 2016
%F A082550 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 A082550 a(n) = A309402(n,n). - _Alois P. Heinz_, Jul 28 2019
%e A082550 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 A082550 Table[Length[Select[Select[Subsets[Range[n]],Max[#]==n&], IntegerQ[ Mean[ #]]&]], {n,22}] (* _Harvey P. Dale_, Jul 23 2011 *)
%t A082550 Table[Total[Table[Length[Select[Select[Subsets[Range[n]], Length[#] == k &],IntegerQ[Total[#]/n] &]], {k, n}]], {n, 10}] (* _Dimitri Papadopoulos_, Jan 18 2016 *)
%o A082550 (PARI) a(n) = sumdiv(n, d, (d%2)* 2^(n/d)*eulerphi(d))/n - 1; \\ _Michel Marcus_, Feb 10 2016
%o A082550 (Python)
%o A082550 from sympy import totient, divisors
%o A082550 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 A082550 Cf. A008965, A051293, A063776, A309402.
%Y A082550 Row sums of A267632.
%K A082550 easy,nonn
%O A082550 1,3
%A A082550 _Naohiro Nomoto_, May 03 2003
%E A082550 a(22) from _Harvey P. Dale_, Jul 23 2011
%E A082550 a(23)-a(32) from _Dimitri Papadopoulos_, Jan 18 2016