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

%I A335068 #10 Jan 12 2024 13:07:26
%S A335068 1,2,4,6,12,30,36,60,180,240,420,840,1680,2520,5040,7560,12600,15120,
%T A335068 30240,55440,110880,221760,332640,665280,720720,1441440,2882880,
%U A335068 3603600,4324320,10810800,24504480,36756720,41081040,43243200,64864800,73513440,122522400,183783600
%N A335068 Numbers k where records occur for sigma(k)/sigma(k+1), where sigma(k) is the sum of divisors of k (A000203).
%C A335068 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.
%H A335068 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 A335068 The values of sigma(k)/sigma(k+1) for the first terms are 0.333..., 0.75, 1.166..., 1.5, 2, ...
%t A335068 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
%t A335068 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 *)
%Y A335068 Cf. A000203, A282531, A326393, A335067.
%K A335068 nonn
%O A335068 1,2
%A A335068 _Amiram Eldar_, May 22 2020