A335068 Numbers k where records occur for sigma(k)/sigma(k+1), where sigma(k) is the sum of divisors of k (A000203).

1, 2, 4, 6, 12, 30, 36, 60, 180, 240, 420, 840, 1680, 2520, 5040, 7560, 12600, 15120, 30240, 55440, 110880, 221760, 332640, 665280, 720720, 1441440, 2882880, 3603600, 4324320, 10810800, 24504480, 36756720, 41081040, 43243200, 64864800, 73513440, 122522400, 183783600

Offset: 1

Author

Amiram Eldar, May 22 2020

Keywords

• nonn

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) can be arbitrarily close to 0 and thus sigma(k)/sigma(k+1) is unbounded and this sequence is infinite.

Examples

			The values of sigma(k)/sigma(k+1) for the first terms are 0.333..., 0.75, 1.166..., 1.5, 2, ...
		

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.

Crossrefs

Cf. A000203, A282531, A326393, A335067.

Programs

Mathematica

rm = 0; s1 = 1; seq = {}; Do[s2 = DivisorSigma[1,n]; If[(r = s1/s2) > rm, rm = r; AppendTo[seq, n-1]]; s1 = s2, {n, 2, 10^5}]; seq
With[{nn=721000},DeleteDuplicates[Thread[{Range[nn-1],#[[1]]/#[[2]]&/@Partition[ DivisorSigma[ 1,Range[nn]],2,1]}],GreaterEqual[#1[[2]],#2[[2]]]&]][[;;,1]] (* The program generates the first 25 terms of the sequence. *) (* Harvey P. Dale, Jan 12 2024 *)