A335067 - OEIS

A335067

Numbers k where records occur for sigma(k+1)/sigma(k), where sigma(k) is the sum of divisors of k (A000203).

%I #9 Jan 08 2021 22:04:55

%S 1,179,239,359,719,839,1259,3359,5039,10079,25199,27719,50399,55439,

%T 110879,166319,360359,665279,831599,1081079,1441439,2162159,3603599,

%U 4324319,12972959,21621599,43243199,61261199,73513439,122522399,205405199,245044799,410810399

%N Numbers k where records occur for sigma(k+1)/sigma(k), where sigma(k) is the sum of divisors of k (A000203).

%C Shapiro (1978) proved that the closure of the set {sigma(k+1)/sigma(k) | k >= 1} consists of all the nonnegative reals. In particular, sigma(k+1)/sigma(k) is unbounded and therefore this sequence is infinite.

%C 25199 is the first composite term.

%H Roy E. DeMeo, Jr., <a href="https://www.jstor.org/stable/2319367">Problem 6107</a>, Advanced Problems, The American Mathematical Monthly, Vol. 83, No. 7 (1976), p. 573, <a href="https://www.jstor.org/stable/2321187">The Closure of sigma(n+1)/sigma(n)</a>, solution by Harold N. Shapiro, ibid., Vol. 85, No. 4 (1978), pp. 287-289.

%e The values of sigma(k+1)/sigma(k) for the first terms are 3, 3.033..., 3.1, 3.25, 3.358..., ...

%t rm = 0; s1 = 1; seq = {}; Do[s2 = DivisorSigma[1,n]; If[(r = s2/s1) > rm, rm = r; AppendTo[seq, n-1]]; s1 = s2, {n, 2, 10^5}]; seq

%Y Cf. A000203, A282531, A326393, A335068.

%K nonn

%O 1,2

%A _Amiram Eldar_, May 22 2020