A273999 Numbers of the form n^2+1 that divide 4^n-1.

1, 5, 17, 257, 46657, 65537, 148997, 67371265, 405458497, 1370776577, 3497539601, 4294967297, 80542440001, 422240040001, 1911029760001, 139251776898727937, 286245437364810001, 6017402415698251777, 18446744073709551617

Offset: 1

Author

Jaroslav Krizek, Jun 06 2016

Keywords

• nonn
• more

Comments

Corresponding values of n are given by A273870(k)-1 for k>=1.

Contains Fermat numbers (A000215) greater than 3.

Also, numbers of the form n^2+1 that divide (4^k)^n-1 for all k >= 0.

a(20) > 4*10^24, if it exists. - Giovanni Resta, Feb 26 2020

Examples

			17 = 4^2+1 is a term because divides 4^4-1; 255 / 17 = 15.
		

Crossrefs

Subsequence of A002522 (numbers of the form n^2+1).

Prime terms are in A274000.

Cf. A000215, A002522, A273870, A273871, A274000.

Programs

PARI

is(n) = ceil(sqrt(n-1))==sqrtint(n-1) && Mod(4, n)^(sqrtint(n))==1
for(n=0, 1e12, if(is(n^2+1), print1(n^2+1, ", "))) \\ Felix Fröhlich, Jun 06 2016

Formula

a(n) = (A273870(n)-1)^2+1.

Extensions

a(16)-a(19) from Lars Blomberg, Aug 10 2016

Edited by Max Alekseyev, Apr 30 2018