A270342 Positive integers n such that the sum of the Pell numbers A000129(0) + ... + A000129(n-1) is divisible by n.

3, 4, 5, 7, 8, 11, 13, 16, 17, 19, 23, 24, 29, 31, 32, 37, 41, 43, 47, 48, 53, 59, 61, 64, 67, 71, 72, 73, 79, 83, 89, 96, 97, 101, 103, 107, 109, 113, 120, 127, 128, 131, 137, 139, 144, 149, 151, 157, 163, 167, 168, 169, 173, 179, 181, 191, 192, 193, 197, 199, 211, 216, 223, 227, 229

Offset

1,1

Comments

Sequence contains all odd primes because of the fact that ((1-sqrt(2))^p + (1+sqrt(2))^p - 2) is divisible by p where p is an odd prime.

Links

Harvey P. Dale, Table of n, a(n) for n = 1..1000

Example

3 is a term because 0 + 1 + 2 = 3 is divisible by 3.
4 is a term because 0 + 1 + 2 + 5 = 8 is divisible by 4.
5 is a term because 0 + 1 + 2 + 5 + 12 = 20 is divisible by 5.
7 is a term because 0 + 1 + 2 + 5 + 12 + 20 + 79 = 119 is divisible by 7.

Mathematica

Module[{nn=250, pell}, pell=LinearRecurrence[{2, 1}, {0, 1}, nn]; Position[ Table[ Total[Take[pell, n]]/n, {n, nn}], _?(IntegerQ[#]&)]]//Flatten (* Harvey P. Dale, Nov 11 2021 *)

Prog

(PARI) a048739(n) = local(w=quadgen(8)); -1/2+(3/4+1/2*w)*(1+w)^n+(3/4-1/2*w)*(1-w)^n;
for(n=1, 1e3, if(a048739(n-1) % (n+1) == 0, print1(n+1, ", ")));

Crossrefs

Cf. A000040, A000129, A048739.

Sequence in context: A111801 A108372 A325131 * A066542 A003310 A038525

Adjacent sequences: A270339 A270340 A270341 * A270343 A270344 A270345

Keyword

nonn

Author

Altug Alkan, Mar 15 2016

Status

approved