A273870 -id:A273870 - cp's OEIS frontend

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

Jaroslav Krizek, Jun 06 2016

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

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

Subsequence of A002522 (numbers of the form n^2+1).
Prime terms are in A274000.
Cf. A000215, A002522, A273870, A273871, A274000.

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

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

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

Edited by Max Alekseyev, Apr 30 2018

A273871 Primes p such that (4^(p-1)-1) == 0 mod ((p-1)^2+1).

Original entry on oeis.org

3, 5, 17, 257, 8209, 59141, 65537, 649801

Offset: 1

Jaroslav Krizek, Jun 01 2016

Prime terms from A273870.
The first 5 known Fermat primes from A019434 are in this sequence.
Conjecture 1: also primes p such that ((4^k)^(p-1)-1) == 0 mod ((p-1)^2+1) for all k >= 0.
Conjecture 2: supersequence of Fermat primes (A019434).

			5 is a term because (4^(5-1)-1) == 0 mod ((5-1)^2+1); 255 == 0 mod 17.

Cf. A019434, A273870.

[n: n in [1..100000] | IsPrime(n) and (4^(n-1)-1) mod ((n-1)^2+1) eq 0];

is(n)=isprime(n) && Mod(4,(n-1)^2+1)^(n-1)==1 \\ Charles R Greathouse IV, Jun 08 2016

cp's OEIS Frontend

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

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