A286181 -id:A286181 - 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

A286208 Greater of Wilson's pseudo-twin primes: primes q such that p! == 1 (mod q), where p=A151799(q) is the previous prime before q, and q-p>2.

Original entry on oeis.org

7853, 594278556271609021, 4259842839142238791410741595983041626644087509

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 A286181 concern consecutive primes (p,q) satisfying p! = 1 (mod q), where d = q-p > 2.
It follows that (d-1)! == 1 (mod q). Listed terms correspond to d = 12, 30, 76 (cf. A286230). Further terms should have d>=140.
Also, primes q=prime(n) such that A275111(n-1)=1, and (prime(n-1),prime(n)) are not twin primes (i.e., q is not a term of A006512).

Subsequence of A166864.

Cf. A275111, A286181, A286230.

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.

A096292 Primes p such that p!-1 is divisible by the next prime larger than p.

Original entry on oeis.org

3, 5, 11, 17, 29, 41, 59, 71, 101, 107, 137, 149, 179, 191, 197, 227, 239, 269, 281, 311, 347, 419, 431, 461, 521, 569, 599, 617, 641, 659, 809, 821, 827, 857, 881, 1019, 1031, 1049, 1061, 1091, 1151, 1229, 1277, 1289, 1301, 1319, 1427, 1451, 1481, 1487, 1607

Offset: 1

Stuart Gascoigne (stuart.g(AT)scoigne.com), Jun 24 2004

nonn

This sequence includes A001359 as a subset, since the lesser of twin primes always satisfy this relation. The smallest number in this sequence that is not in A001359 is 7841. The only other known differences are 594278556271608991 and 4259842839142238791410741595983041626644087433.

			17 is here because 17!-1 is divisible by 19 and 19 is the next prime larger than 17.

Jeppe Stig Nielsen, Table of n, a(n) for n = 1..600
Carlos Rivera, Puzzle 270.

Cf. A001359, A286181.

p=2; forprime(q=3, , prod(j=1, p, Mod(j, q)) == 1 && print1(p, ", "); p=q) \\ Jeppe Stig Nielsen, Oct 04 2019

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

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A286208 Greater of Wilson's pseudo-twin primes: primes q such that p! == 1 (mod q), where p=A151799(q) is the previous prime before q, 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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A096292 Primes p such that p!-1 is divisible by the next prime larger than p.

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PARI