

A270342


Positive integers n such that the sum of the Pell numbers A000129(0) + ... + A000129(n1) is divisible by n.


6



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
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OFFSET

1,1


COMMENTS

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


LINKS



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+(3/41/2*w)*(1w)^n;
for(n=1, 1e3, if(a048739(n1) % (n+1) == 0, print1(n+1, ", ")));


CROSSREFS



KEYWORD

nonn


AUTHOR



STATUS

approved



