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
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
KEYWORD
nonn
AUTHOR
Amiram Eldar, May 22 2020
STATUS
approved