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%I #44 May 05 2021 20:55:00
%S 0,1,2,4,6,10,12,16,18,20,22,28,30,36,40,42,44,46,48,52,58,60,66,70,
%T 72,78,80,82,84,88,90,92,96,100,102,106,108,110,112,116,120,126,130,
%U 132,136,138,140,144,148,150,156,162,164,166,172,174,176,178,180,190,192
%N Rank of the n-th distinct value of the Bernoulli denominators in the sequence of the denominators of the Bernoulli numbers.
%C 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.
%C 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
%C For n > 0, numbers n such that A002322(A027642(n)) = n. - _Thomas Ordowski_, Jul 11 2018
%C Carl Pomerance (in answer to my question) proved that the set of these numbers has asymptotic density zero. - _Thomas Ordowski_, Apr 28 2021
%e 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.
%t 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 *)
%o (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
%Y Cf. A002322, A002445, A027642, A090126, A090801, A317210.
%K nonn
%O 0,3
%A _Paul Curtz_, Oct 09 2014