%I #26 May 06 2016 00:18:19

%S 3,7,9,11,19,23,31,43,47,67,71,83,107,127,131,139,151,163,167,191,211,

%T 263,271,283,307,311,331,347,359,367,383,431,439,463,467,479,491,499,

%U 503,523,547,563,571,587,619,631,647,659,691,719,727,739,743,787,811,823,839,859,863,883,887

%N Numbers n such that A256832(n)/A000129(n-1) is not divisible by n.

%C The computation of integers n such that A256832(n) is not divisible by n, leads to A213891. This sequence contains A213891 as a subsequence.

%C It appears that 9 is the only composite number in this sequence.

%C No composites below 10^7. - _Charles R Greathouse IV_, Apr 20 2016

%C No composites below 2*10^7. - _Charles R Greathouse IV_, May 06 2016

%H Charles R Greathouse IV, <a href="/A270834/b270834.txt">Table of n, a(n) for n = 1..10000</a>

%e 7 is a term because 1*2*5*12*29*169 = 588120 is not divisible by 7.

%t With[{s = Sqrt@ 2}, Select[Range[2, 90], ! Divisible[Product[Expand[((1 + s)^k - (1 - s)^k)/2 s], {k, #}]/Simplify[((1 + s)^(# - 1) - (1 - s)^(# -

%t 1))/(2 s)], #] &]] (* _Michael De Vlieger_, Mar 24 2016, after _Vaclav Kotesovec_ at A256832 and _Michael Somos_ at A000129 *)

%o (PARI) a000129(n) = ([2, 1; 1, 0]^n)[2, 1];

%o t(n) = Mod((prod(k=1, n, a000129(k)) / a000129(n-1)), n);

%o for(n=2, 1e3, if(lift(t(n)) != 0, print1(n, ", ")));

%o (PARI) is(n)=my(a,b=Mod(1,n),t=b); for(k=2,n-2,[a,b]=[b,a+2*b]; t*=b; if(t==0, return(0))); t*(2*a+5*b) && n>2 \\ _Charles R Greathouse IV_, Mar 24 2016

%Y Cf. A000129, A213891, A256832, A270617.

%K nonn

%O 1,1

%A _Altug Alkan_, Mar 23 2016