A274000 -id:A274000 - 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

A274002 Primes p of the form (q-1)^2+1 that are a divisor of 4^(q-1)-1 where q is prime.

Original entry on oeis.org

5, 17, 257, 65537, 3497539601

Offset: 1

Jaroslav Krizek, Jun 06 2016

Corresponding values of primes q: 3, 5, 17, 257, 59141, ...
The first 4 known Fermat primes > 3 from A019434 are in this sequence.
Conjecture: also primes p of the form (q-1)^2+1, where q = prime, that are a divisor of (4^k)^(q-1)-1 for all k>=0. Example: 17 = (5-1)^2+1 is a term because 5 is prime and divides (4^k)^(5-1)-1 for all k>=0: 0/17 = 0 (k=0); 255/17 = 15 (k=1); 65535/17 = 3855 (k=2); 16777215/17 = 986895 (k=3); 4294967295/17 = 252645135 (k=4); 1099511627775/17 = 64677154575 (k=5); ...
Subsequence of A274000.

			17 = (5-1)^2+1 is a term because 17 divides 4^(5-1)-1; 255/17 = 15.

Cf. A019434, A273999, A274000.

listp(nn) = {forprime(p=2, nn, if (isprime(q=(p-1)^2 + 1) &&  (Mod(4, q)^(p-1) == 1), print1(q, ", ")););} \\ Michel Marcus, Jun 08 2016

cp's OEIS Frontend

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

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