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A082550
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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.
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23
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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
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OFFSET
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1,3
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COMMENTS
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Equivalently, number of nonempty subsets of [n] the sum of whose elements is divisible by n. - Dimitri Papadopoulos, Jan 18 2016
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LINKS
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FORMULA
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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
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EXAMPLE
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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.
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MATHEMATICA
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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 *)
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PROG
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(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
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CROSSREFS
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KEYWORD
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easy,nonn
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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