A286181 Lesser of Wilson's pseudo-twin primes: primes p such that p! == 1 (mod q), where q=A151800(p) is the next prime after p, and q-p>2.

7841, 594278556271608991, 4259842839142238791410741595983041626644087433

Offset: 1

Author

Max Alekseyev and Thomas Ordowski, May 04 2017

Keywords

• bref
• nonn
• more

Comments

By Wilson's theorem, p! == 1 (mod p+2) whenever p,p+2 are twin primes. This sequence and A286208 concern consecutive primes p,q satisfying p! = 1 (mod q), where d = q-p > 2.

It follows that (d-1)! == 1 (mod q), and so q divides A033312(d-1).

Listed terms correspond to d = 12, 30, 76 (cf. A286230). Further terms should have d>=140.

Also, primes p=prime(n) such that A275111(n)=1, and (prime(n),prime(n+1)) are not twin primes (i.e., p is not a term of A001359).

Examples

			For a(1)=7841, we have 7841! == 1 (mod 7853), where 7841 and 7853=7841+12 are consecutive primes. Also, 7853 | (12-1)!-1.
		

Crossrefs

Cf. A275111, A286208.