A335067 Numbers k where records occur for sigma(k+1)/sigma(k), where sigma(k) is the sum of divisors of k (A000203).

1, 179, 239, 359, 719, 839, 1259, 3359, 5039, 10079, 25199, 27719, 50399, 55439, 110879, 166319, 360359, 665279, 831599, 1081079, 1441439, 2162159, 3603599, 4324319, 12972959, 21621599, 43243199, 61261199, 73513439, 122522399, 205405199, 245044799, 410810399

Offset

1,2

Comments

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.

25199 is the first composite term.

Links

Table of n, a(n) for n=1..33.

Roy E. DeMeo, Jr., Problem 6107, Advanced Problems, The American Mathematical Monthly, Vol. 83, No. 7 (1976), p. 573, The Closure of sigma(n+1)/sigma(n), solution by Harold N. Shapiro, ibid., Vol. 85, No. 4 (1978), pp. 287-289.

Example

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

Mathematica

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

Crossrefs

Cf. A000203, A282531, A326393, A335068.

Sequence in context: A162164 A238893 A230809 * A142028 A146359 A142441

Adjacent sequences: A335064 A335065 A335066 * A335068 A335069 A335070

Keyword

nonn

Author

Amiram Eldar, May 22 2020

Status

approved