A330245 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).

6, 20, 78, 1014, 3774, 9514254

Offset: 1

Author

Amiram Eldar, Dec 06 2019

Keywords

• nonn
• more

Comments

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.

10^11 < a(7) <= 105590246974194. - Giovanni Resta, Jan 14 2020

Examples

			The abundancy indices of the terms are sigma(a(n))/a(n) = 2 < 2.1 < 2.153... < 2.165... < 2.174... < 2.1757...
		

References

• Richard K. Guy, Unsolved Problems in Number Theory, 3rd Edition, Springer, 2004, Section B2, p. 77.

Crossrefs

Cf. A000203, A064771, A068403.

Programs

Mathematica

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 *)

Extensions

a(6) from Giovanni Resta, Jan 14 2020