cp's OEIS Frontend

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

%I A270342 #23 Nov 11 2021 13:35:54
%S A270342 3,4,5,7,8,11,13,16,17,19,23,24,29,31,32,37,41,43,47,48,53,59,61,64,
%T A270342 67,71,72,73,79,83,89,96,97,101,103,107,109,113,120,127,128,131,137,
%U A270342 139,144,149,151,157,163,167,168,169,173,179,181,191,192,193,197,199,211,216,223,227,229
%N A270342 Positive integers n such that the sum of the Pell numbers A000129(0) + ... + A000129(n-1) is divisible by n.
%C A270342 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.
%H A270342 Harvey P. Dale, <a href="/A270342/b270342.txt">Table of n, a(n) for n = 1..1000</a>
%e A270342 3 is a term because 0 + 1 + 2 = 3 is divisible by 3.
%e A270342 4 is a term because 0 + 1 + 2 + 5 = 8 is divisible by 4.
%e A270342 5 is a term because 0 + 1 + 2 + 5 + 12 = 20 is divisible by 5.
%e A270342 7 is a term because 0 + 1 + 2 + 5 + 12 + 20 + 79 = 119 is divisible by 7.
%t A270342 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 *)
%o A270342 (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;
%o A270342 for(n=1, 1e3, if(a048739(n-1) % (n+1) == 0, print1(n+1, ", ")));
%Y A270342 Cf. A000040, A000129, A048739.
%K A270342 nonn
%O A270342 1,1
%A A270342 _Altug Alkan_, Mar 15 2016