A282531 Numbers k where records occur for d(k+1)/d(k), where d(k) is A000005(k).

1, 11, 23, 47, 59, 167, 179, 239, 359, 719, 839, 1259, 2879, 3359, 5039, 7559, 10079, 21839, 33599, 35279, 37799, 55439, 100799, 110879, 166319, 262079, 327599, 415799, 665279, 831599, 1081079, 1441439, 2827439, 3326399, 4989599, 6320159, 6486479, 10533599

Offset: 1

Author

Daniel Suteu, Feb 18 2017

Keywords

• nonn

Comments

This sequence is infinite (Schinzel, 1954). - Amiram Eldar, Apr 18 2024

Links

• Amiram Eldar, Table of n, a(n) for n = 1..71 (terms 1..52 from Daniel Suteu)
• Andrzej Schinzel, Sur une propriété du nombre de diviseurs, Publ. Math. (Debrecen), Vol. 3 (1954), pp. 261-262.

Crossrefs

Cf. A000005, A372092.

Programs

Mathematica

seq[kmax_] := Module[{d1 = 1, d2, rm = 0, r, s = {}}, Do[d2 = DivisorSigma[0, k]; r = d2 / d1; If[r > rm, rm = r; AppendTo[s, k-1]]; d1 = d2, {k, 2, kmax}]; s]; seq[10^6] (* Amiram Eldar, Apr 18 2024 *)
Module[{nn=840000},DeleteDuplicates[Thread[{Range[nn-1],#[[2]]/#[[1]]&/@Partition[ DivisorSigma[ 0,Range[nn]],2,1]}],GreaterEqual[#1[[2]],#2[[2]]]&]][[;;,1]] (* The program generates the first 30 terms of the sequence. *) (* Harvey P. Dale, Jun 10 2024 *)

PARI

lista(kmax) = {my(d1 = 1, d2, rm = 0, r); for(k = 2, kmax, d2 = numdiv(k); r = d2 / d1; if(r > rm, rm = r; print1(k-1, ", ")); d1 = d2);} \\ Amiram Eldar, Apr 18 2024

Perl

use ntheory qw(:all);
for (my ($n, $m) = (1, 0) ; ; ++$n) {
    my $d = divisors($n+1) / divisors($n);
    if ($m < $d) {
        $m = $d;
        print "$n\n";
    }
}