A286208 -id:A286208 - cp's OEIS frontend

A275111 a(n) = prime(n)! mod prime(n+1).

2, 1, 1, 2, 1, 3, 1, 4, 22, 1, 33, 7, 1, 8, 19, 30, 1, 43, 12, 1, 27, 14, 23, 24, 17, 1, 18, 1, 19, 19, 22, 8, 1, 94, 1, 140, 72, 28, 62, 91, 1, 105, 1, 33, 1, 177, 97, 38, 1, 39, 2, 1, 19, 15, 160, 204, 1, 247, 47, 1, 291, 299, 52, 1, 53, 198, 132, 55, 1, 59, 3, 176

Offset: 1

Thomas Ordowski, Jul 17 2016

nonn

By Wilson's theorem, if prime(n+1) - prime(n) = 2 then a(n) = 1.

However a(991) = 1, while prime(992) - prime(991) = 7853 - 7841 = 12. See A286181, A286208, A286230. - Robert Israel, Jul 17 2016

Chai Wah Wu, Table of n, a(n) for n = 1..10000

Cf. A286181, A286208, A286230.

Table[Mod[#!, NextPrime@ #] &@ Prime@ n, {n, 120}] (* Michael De Vlieger, Jul 17 2016 *)

a(n) = prime(n)! % prime(n+1); \\ Michel Marcus, Jul 17 2016

a(n,p=prime(n))=my(q=nextprime(p+1)); if(p==2, 2, lift( 1/prod(r=p+1,q-2, Mod(r,q)) ) ); \\ Charles R Greathouse IV, Jul 18 2016; corrected by Max Alekseyev, May 03 2017

a(n,p=prime(n)) = my(q=nextprime(p+1)); if(p==2, 2, (1/(q-p-1)!)%q); \\ Max Alekseyev, May 03 2017

from sympy import prime
from sympy.core.numbers import igcdex
def A275111(n):
    p, q = prime(n), prime(n+1)
    a = q-1
    for i in range(p+1,q):
        a = (a*igcdex(i,q)[0]) % q
    return a # Chai Wah Wu, Jul 18 2016

from functools import reduce
from sympy import prime
def A275111(n): return ((q:=prime(n+1))-1)*pow(reduce(lambda i,j:i*j%q,range(prime(n)+1,q),1),-1,q)%q # Chai Wah Wu, Feb 24 2023

For n>1, a(n) = 1/((prime(n)+1)*(prime(n)+2)*...*(prime(n+1)-2)) mod prime(n+1). - Robert Israel, Jul 17 2016; corrected by Max Alekseyev, May 03 2017

For n>1, a(n) = 1/(prime(n+1)-prime(n)-1)! mod prime(n+1) = 1/(A001223(n)-1)! mod A000040(n+1). - Max Alekseyev, May 03 2017

More terms from Altug Alkan, Jul 17 2016

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.

Original entry on oeis.org

7841, 594278556271608991, 4259842839142238791410741595983041626644087433

Offset: 1

Max Alekseyev and Thomas Ordowski, May 04 2017

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).

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

Cf. A275111, A286208.

A286230 Possible differences between consecutive primes p, q satisfying p! == 1 (mod q).

Original entry on oeis.org

2, 12, 30, 76

Offset: 1

Max Alekseyev, May 04 2017

By Wilson's theorem, p! == 1 (mod p+2) whenever p, p+2 are twin primes, so 2 is a term.
Terms d > 2 correspond to Wilson's pseudo-twin primes, i.e., consecutive primes p, q such that q - p = d and p! == 1 (mod q). The corresponding primes are listed in A286181 and A286208.
As a set, the sequence is the union of {2} and the differences {A286208(n) - A286181(n)}.
A positive integer d belongs to this sequence if A033312(d-1) = (d-1)! - 1 has a prime factor q such that q - A151799(q) = d.
All terms are even.
If it exists, a(5) >= 140.

Cf. A275111, A286181, A286208.

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A275111 a(n) = prime(n)! mod prime(n+1).

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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.

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A286230 Possible differences between consecutive primes p, q satisfying p! == 1 (mod q).

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