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

Author

Altug Alkan, Mar 15 2016

Keywords

• nonn

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.

Examples

			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.
		

Links

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

Crossrefs

Cf. A000040, A000129, A048739.

Programs

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 *)

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, ", ")));