A248614 Rank of the n-th distinct value of the Bernoulli denominators in the sequence of the denominators of the Bernoulli numbers.

0, 1, 2, 4, 6, 10, 12, 16, 18, 20, 22, 28, 30, 36, 40, 42, 44, 46, 48, 52, 58, 60, 66, 70, 72, 78, 80, 82, 84, 88, 90, 92, 96, 100, 102, 106, 108, 110, 112, 116, 120, 126, 130, 132, 136, 138, 140, 144, 148, 150, 156, 162, 164, 166, 172, 174, 176, 178, 180, 190, 192

Offset: 0

Author

Paul Curtz, Oct 09 2014

Keywords

• nonn

Comments

Consider sequence A027642 of the denominators of the Bernoulli numbers and the reduced sequence b(n) = 1, 2, 6, 30, 42, 66,... if duplicates are removed (which is 1, 2 followed by A090126). a(n) shows the smallest index --place of first appearance-- of b(n) in the full list A027642.

If n is of the form A002322(p*q) with p*q semiprime, then n is a term. The number 3652 is a term, but it is not of the form A002322(p*q), as Carl Pomerance noted. - Thomas Ordowski, Apr 28 2021; in place of an incorrect comment by Filip Zaludek, Sep 23 2016

For n > 0, numbers n such that A002322(A027642(n)) = n. - Thomas Ordowski, Jul 11 2018

Carl Pomerance (in answer to my question) proved that the set of these numbers has asymptotic density zero. - Thomas Ordowski, Apr 28 2021

Examples

			b(2)=6 appears first in A027642(2), so a(2)=2. b(4)=42 appears first as A027642(6)=42, so a(4)=6. b(5)=66 appears first as A027642(10), so a(5)=10.
		

Crossrefs

Cf. A002322, A002445, A027642, A090126, A090801, A317210.

Programs

Mathematica

BB = Table[Denominator[BernoulliB[n]], {n, 2, 400, 2}]; For[t = BB; n = 1, n <= Length[t], n++, p = Position[t, t[[n]]] // Rest; t = Delete[t, p]]; reducedBB = Join[{1, 2}, t]; a[0] = 0; a[1] = 1; a[n_] := 2*Position[BB, reducedBB[[n+1]], 1, 1][[1, 1]]; Table[a[n], {n, 0, 100}] (* Jean-François Alcover, Oct 16 2014 *)

PARI

L=List(); N=60; forprime(p=2, N*N, forprime(q=p, N*N, listput(L, lcm(p-1,q-1)) )); listsort(L, 1); for (i=1, N, print1(L[i], ", ")) \\ Filip Zaludek, Sep 23 2016