cp's OEIS Frontend

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).

%I A333054 #8 Jan 02 2021 04:50:12
%S A333054 1,2,8,14,44,104,495,944,4095,5775,5984,21735,98175,862784,4096575,
%T A333054 7194824,14753024,879207615,1969789184,2275962975,3968862975,
%U A333054 12567844575,39566665215,44803620225,77510285775,125617830975,162902829375
%N 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).
%C A333054 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...
%C A333054 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).
%C A333054 a(28) > 5*10^11. - _Amiram Eldar_, Jan 02 2021
%e A333054 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.
%t A333054 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
%Y A333054 Cf. A000203, A004394, A068403, A096399, A333053.
%K A333054 nonn,more
%O A333054 1,2
%A A333054 _Amiram Eldar_, Mar 06 2020
%E A333054 a(22)-a(27) from _Amiram Eldar_, Jan 02 2021