A282531 -id:A282531 - cp's OEIS frontend

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

Amiram Eldar, May 22 2020

nonn

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.

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

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.

Cf. A000203, A282531, A326393, A335068.

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

A372092 Numbers k where records occur for d(k)/d(k+1), where d(k) is the number of divisors of k (A000005).

Original entry on oeis.org

1, 2, 4, 6, 12, 30, 36, 60, 180, 240, 420, 1008, 1320, 1800, 2160, 2520, 6300, 7560, 12600, 15120, 20160, 30240, 45360, 55440, 100800, 110880, 196560, 332640, 498960, 786240, 982800, 1108800, 1580040, 1940400, 1995840, 2402400, 3880800, 4324320, 11476080, 11531520

Offset: 1

Amiram Eldar, Apr 18 2024

nonn

This sequence is infinite (Schinzel, 1954).
Is a(n) = A103199(n) - 1?
From Michael De Vlieger, Apr 19 2024: (Start)
a(12) = 1008 = 2^4 * 3^2 * 7 is the smallest term that is not a product of primorials.
a(36) = 2402400 = 2^5 * 3^1 * 5^2 * 7 * 11 * 13 is the smallest term whose exponents are not nonincreasing as prime base increases (ignoring interposing nondivisor primes). (End)

Amiram Eldar, Table of n, a(n) for n = 1..69
Michael De Vlieger, Prime power decomposition of a(n), n = 1..69.
Andrzej Schinzel, Sur une propriété du nombre de diviseurs, Publ. Math. (Debrecen), Vol. 3 (1954), pp. 261-262.

Cf. A000005, A103199, A282531.

seq[kmax_] := Module[{d1 = 1, d2, rm = 0, r, s = {}}, Do[d2 = DivisorSigma[0, k]; r = d1 / d2; If[r > rm, rm = r; AppendTo[s, k-1]]; d1 = d2, {k, 2, kmax}]; s]; seq[10^6]

lista(kmax) = {my(d1 = 1, d2, rm = 0, r); for(k = 2, kmax, d2 = numdiv(k); r = d1 / d2; if(r > rm, rm = r; print1(k-1, ", ")); d1 = d2);}

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

Original entry on oeis.org

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

Amiram Eldar, May 22 2020

nonn

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.

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

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.

Cf. A000203, A282531, A326393, A335067.

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 *)

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

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A372092 Numbers k where records occur for d(k)/d(k+1), where d(k) is the number of divisors of k (A000005).

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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).

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Mathematica