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%I #20 Jan 14 2020 09:37:46
%S 6,20,78,1014,3774,9514254
%N 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).
%C 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.
%C a(6) > 2*10^5.
%C 10^11 < a(7) <= 105590246974194. - _Giovanni Resta_, Jan 14 2020
%D Richard K. Guy, Unsolved Problems in Number Theory, 3rd Edition, Springer, 2004, Section B2, p. 77.
%e The abundancy indices of the terms are sigma(a(n))/a(n) = 2 < 2.1 < 2.153... < 2.165... < 2.174... < 2.1757...
%t 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 *)
%Y Cf. A000203, A064771, A068403.
%K nonn,more
%O 1,1
%A _Amiram Eldar_, Dec 06 2019
%E a(6) from _Giovanni Resta_, Jan 14 2020
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