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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).

%I A372092 #14 Apr 20 2024 10:49:49
%S A372092 1,2,4,6,12,30,36,60,180,240,420,1008,1320,1800,2160,2520,6300,7560,
%T A372092 12600,15120,20160,30240,45360,55440,100800,110880,196560,332640,
%U A372092 498960,786240,982800,1108800,1580040,1940400,1995840,2402400,3880800,4324320,11476080,11531520
%N A372092 Numbers k where records occur for d(k)/d(k+1), where d(k) is the number of divisors of k (A000005).
%C A372092 This sequence is infinite (Schinzel, 1954).
%C A372092 Is a(n) = A103199(n) - 1?
%C A372092 From _Michael De Vlieger_, Apr 19 2024: (Start)
%C A372092 a(12) = 1008 = 2^4 * 3^2 * 7 is the smallest term that is not a product of primorials.
%C A372092 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)
%H A372092 Amiram Eldar, <a href="/A372092/b372092.txt">Table of n, a(n) for n = 1..69</a>
%H A372092 Michael De Vlieger, <a href="/A372092/a372092.txt">Prime power decomposition of a(n)</a>, n = 1..69.
%H A372092 Andrzej Schinzel, <a href="https://doi.org/10.5486/PMD.1954.3.3-4.11">Sur une propriété du nombre de diviseurs</a>, Publ. Math. (Debrecen), Vol. 3 (1954), pp. 261-262.
%t A372092 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]
%o A372092 (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);}
%Y A372092 Cf. A000005, A103199, A282531.
%K A372092 nonn
%O A372092 1,2
%A A372092 _Amiram Eldar_, Apr 18 2024