A333054 Numbers m such that r(m) > r(k) for all k < m, where r(m) = min(sigma(m)/m, sigma(m+1)/(m+1)), and sigma(m) is the sum of divisors of m (A000203).

1, 2, 8, 14, 44, 104, 495, 944, 4095, 5775, 5984, 21735, 98175, 862784, 4096575, 7194824, 14753024, 879207615, 1969789184, 2275962975, 3968862975, 12567844575, 39566665215, 44803620225, 77510285775, 125617830975, 162902829375

Offset

1,2

Comments

The corresponding values of r(a(n)) are 1, 1.333..., 1.444..., 1.6, 1.733..., 1.828..., 1.890..., 1.970..., 1.999..., 2.044..., 2.085..., 2.120..., 2.181..., 2.243..., 2.248..., 2.252..., 2.360..., 2.397..., 2.407..., 2.408..., 2.411...

The least number m such that both m and m+1 are k-abundant (i.e., their abundancy indices sigma(m)/m > k and sigma(m+1)/(m+1) > k) is a term in this sequence. E.g., a(10) = 5775 = A096399(1).

a(28) > 5*10^11. - Amiram Eldar, Jan 02 2021

Links

Table of n, a(n) for n=1..27.

Example

The values of min(sigma(k)/k, sigma(k+1)/(k+1)) for k = 1, 2, ... 8 are 1, 4/3, 4/3, 6/5, 6/5, 8/7, 8/7, 13/9. The record values in this range, 1, 4/3 and 13/9, are obtained at k = 1, 2, and 8.

Mathematica

seq={}; rminmax = 0; r1 = 1; Do[r2 = DivisorSigma[1, n]/n; rmin = Min[r1, r2]; If[rmin > rminmax, rminmax = rmin; AppendTo[seq, n-1]]; r1 = r2, {n, 2, 10^6}]; seq

Crossrefs

Cf. A000203, A004394, A068403, A096399, A333053.

Sequence in context: A259119 A039660 A333053 * A119752 A111001 A055258

Adjacent sequences: A333051 A333052 A333053 * A333055 A333056 A333057

Keyword

nonn,more

Author

Amiram Eldar, Mar 06 2020

Extensions

a(22)-a(27) from Amiram Eldar, Jan 02 2021

Status

approved