A247220 - OEIS

%I #52 Jun 27 2024 22:21:49

%S 0,2,4,386,20136,59140,373164544

%N Numbers k such that k^2 + 1 divides 2^k + 1.

%C a(8) > 10^12. - _Giovanni Resta_, May 05 2020

%C All terms of the sequence are even. a(5), a(6) and a(7) are of the form 2*p + 2 where p is a prime and p mod 14 = 1. - _Farideh Firoozbakht_, Dec 07 2014

%C From _Jianing Song_, Jan 13 2019: (Start)

%C Among the known terms only a(3) and a(4) are of the form 2*p where p is a prime.

%C a(n)^2 + 1 is prime for 2 <= n <= 7. Among these primes, the multiplicative order of 2 modulo a(n)^2 + 1 is 2*a(n) except for n = 5, in which case it is 2*a(n)/3. (End)

%C If a(n)^2 + 1 is composite, then a(n) is also a term of A135590. - _Max Alekseyev_, Apr 25 2024

%e 0 is in this sequence because 0^2 + 1 = 1 divides 2^0 + 1 = 2.

%o (PARI) for(n=0,10^5,if(Mod(2,n^2+1)^n==-1,print1(n,", "))); \\ _Joerg Arndt_, Nov 30 2014

%o (Python)

%o from gmpy2 import powmod

%o A247220_list = [i for i in range(10**7) if powmod(2,i,i*i+1) == i*i]

%o # _Chai Wah Wu_, Dec 03 2014

%Y Cf. A135590, A247165.

%K nonn,more

%O 1,2

%A _Juri-Stepan Gerasimov_, Nov 26 2014

%E a(7) from _Hiroaki Yamanouchi_, Nov 29 2014