login
Numbers k such that Pell(k) and Pell(k+1) have the same prime signature where Pell(k) = A000129(k).
0

%I #26 Jul 07 2024 16:47:58

%S 2,14,26,38,86,94,106,214,218,334,338,381,590,605

%N Numbers k such that Pell(k) and Pell(k+1) have the same prime signature where Pell(k) = A000129(k).

%C Initial terms are 2, 2*7, 2*13, 2*19, 2*43, 2*47, 2*53, 2*107, 2*109.

%H <a href="/index/Pri#prime_signature">Index to sequences related to prime signature</a>.

%e 14 is a term because A000129(14) = 2 * 13^2 * 239 and A000129(15) = 5^2 * 29 * 269.

%t psig[n_]:=Sort[FactorInteger[n][[;;,2]]]; Drop[With[{nn=220},SequencePosition[psig/@ LinearRecurrence[ {2,1},{0,1},nn],{x_,x_}]][[;;,1]]-1,2] (* The program generates the first 9 terms of the sequence. *) (* _Harvey P. Dale_, Jul 07 2024 *)

%o (PARI) Pell(n) = ([2, 1; 1, 0]^n)[2, 1];

%o is(n) = vecsort(factor(Pell(n))[, 2]) == vecsort(factor(Pell(n+1))[, 2]);

%Y Cf. A000129, A100484.

%K nonn,more

%O 1,1

%A _Altug Alkan_, Dec 31 2016

%E a(10)-a(14) from _Amiram Eldar_, May 19 2024