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A330245
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Numbers m with a unique subset of the divisors of m that sums to m (A064771) such that sigma(m)/m > sigma(k)/k for all smaller terms k < m of A064771, where sigma(m) is the sum of divisors of m (A000203).
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0
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OFFSET
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1,1
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COMMENTS
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Paul Erdős asked whether there are extra-weird numbers n, i.e., numbers n for which sigma(n)/n > 3, but n is not the sum of a subset of its divisors in two ways. Such numbers, if they exist, are in the intersection of A064771 and A068403, and the least of them is a term of this sequence.
a(6) > 2*10^5.
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REFERENCES
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Richard K. Guy, Unsolved Problems in Number Theory, 3rd Edition, Springer, 2004, Section B2, p. 77.
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LINKS
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EXAMPLE
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The abundancy indices of the terms are sigma(a(n))/a(n) = 2 < 2.1 < 2.153... < 2.165... < 2.174... < 2.1757...
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MATHEMATICA
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okQ[n_] := Module[{d = Most[Divisors[n]]}, SeriesCoefficient[Series[ Product[ 1+x^i, {i, d}], {x, 0, n}], n] == 1]; seq = {}; rm = 0; Do[If[(r = DivisorSigma[1, n]/n) > rm && okQ[n], rm = r; AppendTo[seq, n]], {n, 1, 4000}]; seq (* after Harvey P. Dale at A064771 *)
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CROSSREFS
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KEYWORD
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nonn,more
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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