

A214028


Entry points for the Pell sequence: smallest k such that n divides A000129(k).


19



1, 2, 4, 4, 3, 4, 6, 8, 12, 6, 12, 4, 7, 6, 12, 16, 8, 12, 20, 12, 12, 12, 22, 8, 15, 14, 36, 12, 5, 12, 30, 32, 12, 8, 6, 12, 19, 20, 28, 24, 10, 12, 44, 12, 12, 22, 46, 16, 42, 30, 8, 28, 27, 36, 12, 24, 20, 10, 20, 12, 31, 30, 12, 64, 21, 12, 68, 8, 44, 6, 70, 24, 36, 38
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OFFSET

1,2


COMMENTS

Conjecture: A175181(n)/A214027(n) = a(n). This says that the zeros appear somewhat uniformly in a period. The second zero in a period is exactly where n divides the first Lucas number, so this relationship is not really surprising.
From Jianing Song, Aug 29 2018: (Start)
The comment above is correct, since n divides A000129(k*a(n)) for all integers k and clearly a(n) divides A175181(n), so the zeros appear uniformly.
a(n) <= 4*n/3 for all n, with equality holds iff n is a power of 3.
(End)


LINKS

G. C. Greubel, Table of n, a(n) for n = 1..10000
Bernadette Faye, Florian Luca, Pell Numbers whose Euler Function is a Pell Number, arXiv:1508.05714 [math.NT], 2015 (called z(n)).
N. Robbins, On Pell numbers of the form p*x^2, where p is prime, Fib. Quart. 22 (4) (1984) 340348, definition 1.


FORMULA

If p^2 does not divide A000129(a(p)) (that is, p is not in A238736) then a(p^e) = a(p)*p^(e  1). If gcd(m, n) = 1 then a(mn) = lcm(a(m), a(n)).  Jianing Song, Aug 29 2018


EXAMPLE

11 first divides the term A000129(12) = 13860 = 2*3*5*7*11.


MAPLE

A214028 := proc(n)
local a000129, k ;
a000129 := [1, 2, 5] ;
for k do
if modp(a000129[1], n) = 0 then
return k;
end if;
a000129[1] := a000129[2] ;
a000129[2] := a000129[3] ;
a000129[3] := 2*a000129[2]+a000129[1] ;
end do:
end proc:
seq(A214028(n), n=1..40); # R. J. Mathar, May 26 2016


MATHEMATICA

a[n_] := With[{s = Sqrt@ 2}, ((1 + s)^n  (1  s)^n)/(2 s)] // Simplify; Table[k = 1; While[Mod[a[k], n] != 0, k++]; k, {n, 80}] (* Michael De Vlieger, Aug 25 2015, after Michael Somos at A000129 *)
Table[k = 1; While[Mod[Fibonacci[k, 2], n] != 0, k++]; k, {n, 100}] (* G. C. Greubel, Aug 10 2018 *)


PROG

(PARI) pell(n) = polcoeff(Vec(x/(12*xx^2) + O(x^(n+1))), n);
a(n) = {k=1; while (pell(k) % n, k++); k; } \\ Michel Marcus, Aug 25 2015


CROSSREFS

Cf. A001175, A001176, A001177, A214027, A175181.
Sequence in context: A202690 A257978 A193358 * A079533 A072872 A135359
Adjacent sequences: A214025 A214026 A214027 * A214029 A214030 A214031


KEYWORD

nonn


AUTHOR

Art DuPre, Jul 04 2012


STATUS

approved



