A051887 -id:A051887 - cp's OEIS frontend

A051888 a(n) is the smallest prime p such that p*n! + 1 is prime.

2, 2, 2, 2, 3, 2, 3, 3, 7, 3, 3, 5, 2, 3, 13, 7, 31, 5, 2, 7, 17, 67, 41, 3, 13, 3, 43, 17, 97, 7, 29, 109, 3, 71, 5, 2, 7, 41, 3, 59, 3, 11, 29, 7, 107, 67, 79, 3, 743, 149, 163, 2, 211, 2, 19, 71, 73, 23, 37, 113, 149, 67, 41, 617, 107, 37, 107, 283, 113, 19, 239, 107, 73, 97, 5

Offset: 0

Labos Elemer, Dec 15 1999

nonn

Analogous to or subset of A051686; generalization of A005384.

The PFGW program has been used to certify all the primes corresponding to the terms up to a(1000), using a deterministic test which exploits the factorization of a(n) - 1. - Giovanni Resta, May 30 2018

Giovanni Resta, Table of n, a(n) for n = 0..1000

Cf. A005384, A051686, A051886, A051887, A051901, A064983.

Do[k = 1; While[ !PrimeQ[ Prime[k]*n! + 1], k++ ]; Print[ Prime[k]], {n, 1, 75} ]
spp[n_]:=Module[{p=2,nf=n!},While[!PrimeQ[p*nf+1],p=NextPrime[p]];p]; Array[ spp,80,0] (* Harvey P. Dale, May 17 2019 *)

a(n) = {my(p=2); while (!isprime(p*n! + 1), p = nextprime(p+1)); p;} \\ Michel Marcus, May 28 2018

a(n) = (A051901(n)-1)/n!. - Amiram Eldar, Feb 25 2025

More terms from James Sellers, Dec 16 1999

A051886 a(n) is the minimal prime p such that 2^n * p + 1 is prime.

Original entry on oeis.org

2, 2, 3, 2, 7, 3, 3, 2, 3, 23, 13, 29, 3, 5, 7, 2, 37, 53, 3, 11, 7, 11, 37, 71, 73, 5, 7, 17, 13, 23, 3, 239, 43, 113, 163, 59, 3, 89, 349, 5, 97, 3, 73, 11, 67, 101, 19, 101, 61, 23, 7, 17, 7, 233, 127, 5, 541, 29, 103, 71, 31, 53, 109, 179, 163, 71, 3, 929, 31, 23, 193, 101

Offset: 0

Labos Elemer, Dec 15 1999

nonn

The minimal 2^n - Germain primes in order of increasing exponent n.

			The 10th term is 13, the first term in 1024-Germain prime sequence: {13,19,37,79,223,...}. The largest prime was found for 2^79: both 1427 and 604462909807314587353088*1427 + 1 = 862568572295037916152856577 are primes.

Joerg Arndt, Table of n, a(n) for n = 0..1000

Cf. A051686, A005384, A023212, A023228, A051887, A051888, A051900.

Table[p = 2; While[! PrimeQ[2^n*p + 1], p = NextPrime@ p]; p, {n, 0, 71}] (* Michael De Vlieger, Mar 05 2017 *)

P=10^6;
default(primelimit,P);
a(n)={my(N=2^n);forprime(p=2,P,if(isprime(N*p+1),return(p)));}
vector(66,n,a(n))
/* Joerg Arndt, Jun 18 2012 */

a(n) = (A051900(n)-1)/2^n. - Amiram Eldar, Feb 28 2025

Better name by Joerg Arndt, Jun 18 2012

A051902 Minimal primorial safe primes: p and primorial*p + 1 are both primes.

Original entry on oeis.org

5, 13, 61, 421, 4621, 150151, 8678671, 106696591, 2454021571, 71166625531, 401120980261, 170676977100631, 2129751844690471, 562558737261811291, 11682905869181336791, 97767475431570134191, 9613801750771063195351, 234576762718813941966541, 55008250857561869391153631

Offset: 1

Labos Elemer, Dec 16 1999

nonn

In A051888, 13 of the first 25 values are distinct, while here all corresponding min-primorial-safe-primes are different: {2,5,17,11,23,43,19,3,7,61,53,41,47}.

			The first five terms of A051887 are 2, so the first 5 terms have the form 1 + 2*A002110(n): 5, 13, 61, 421, 4621, which are smallest terms in A005385, A051644, A051646, A051648, A051649. The 6th term here is A051651(1) = A051887(6)*A002110(6) + 1 = 5*30030 + 1.

Amiram Eldar, Table of n, a(n) for n = 1..349

Cf. A005385, A002110, A051644, A051646, A051648, A051649.

a[n_] := Module[{p = 2, r =Times @@ Prime[Range[n]]}, While[!PrimeQ[p * r + 1], p = NextPrime[p]]; p * r + 1]; Array[a, 20] (* Amiram Eldar, Feb 25 2025 *)

a(n) = {my(p = 2, r = vecprod(primes(n))); while(!isprime(p * r + 1), p = nextprime(p+1)); p * r + 1;} \\ Amiram Eldar, Feb 25 2025

a(n) = 1 + A002110(n)*A051887(n).

A051800 Numbers k such that 1 plus twice the product of the first k primes is also a prime.

Original entry on oeis.org

1, 2, 3, 4, 5, 11, 18, 23, 26, 30, 80, 120, 148, 220, 395, 776, 884, 977, 3535, 3927

Offset: 1

Labos Elemer, Dec 20 1999

			5 is in the sequence because 2*(2*3*5*7*11) + 1 = 4621 is prime.

2*A002110(n)+1 is prime. Cf. A051887, A051915.

Position[2#+1&/@FoldList[Times,Prime[Range[800]]],?PrimeQ]//Flatten (* _Harvey P. Dale, Oct 09 2018 *)

isok(k) = isprime(1+2*prod(j=1, k, prime(j))); \\ Michel Marcus, May 28 2018

from sympy import isprime, nextprime
def afind(limit):
    p, primorialk = 2, 2
    for k in range(1, limit+1):
        if isprime(2*primorialk + 1):
            print(k, end=", ")
        p = nextprime(p)
        primorialk *= p
afind(400) # Michael S. Branicky, Dec 24 2021

More terms from Harvey P. Dale, Oct 09 2018
a(17)-a(18) from Michael S. Branicky, Dec 24 2021
a(19)-a(20) from Michael S. Branicky, May 30 2023

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A051888 a(n) is the smallest prime p such that p*n! + 1 is prime.

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A051886 a(n) is the minimal prime p such that 2^n * p + 1 is prime.

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A051902 Minimal primorial safe primes: p and primorial*p + 1 are both primes.

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Author

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Comments

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Crossrefs

Programs

Mathematica

PARI

Formula

A051800 Numbers k such that 1 plus twice the product of the first k primes is also a prime.

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Author

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Mathematica

PARI

Python

Extensions