A270342 - OEIS

%I #23 Nov 11 2021 13:35:54

%S 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 67,71,72,73,79,83,89,96,97,101,103,107,109,113,120,127,128,131,137,

%U 139,144,149,151,157,163,167,168,169,173,179,181,191,192,193,197,199,211,216,223,227,229

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

%C 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 Harvey P. Dale, <a href="/A270342/b270342.txt">Table of n, a(n) for n = 1..1000</a>

%e 3 is a term because 0 + 1 + 2 = 3 is divisible by 3.

%e 4 is a term because 0 + 1 + 2 + 5 = 8 is divisible by 4.

%e 5 is a term because 0 + 1 + 2 + 5 + 12 = 20 is divisible by 5.

%e 7 is a term because 0 + 1 + 2 + 5 + 12 + 20 + 79 = 119 is divisible by 7.

%t 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 (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 for(n=1, 1e3, if(a048739(n-1) % (n+1) == 0, print1(n+1, ", ")));

%Y Cf. A000040, A000129, A048739.

%K nonn

%O 1,1

%A _Altug Alkan_, Mar 15 2016