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

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

Author

Amiram Eldar, Apr 18 2024

Keywords

• nonn

Comments

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)

Links

• 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.

Crossrefs

Cf. A000005, A103199, A282531.

Programs

Mathematica

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]

PARI

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);}