A051888 -id:A051888 - cp's OEIS frontend

A051901 Minimal factorial safe-primes: a prime p = a(n) here if (p-1)/n! = A051888(n).

3, 3, 5, 13, 73, 241, 2161, 15121, 282241, 1088641, 10886401, 199584001, 958003201, 18681062401, 1133317785601, 9153720576001, 648606486528001, 1778437140480001, 12804747411456001, 851515702861824001, 41359334139002880001, 3423093125504532480001, 46084029838881914880001

Offset: 0

Labos Elemer, Dec 16 1999

nonn

			a(8) = 282241 = 8!*A051888(8) = 40320*7 + 1.

Amiram Eldar, Table of n, a(n) for n = 0..448

Cf. A005385, A051888, A000142.

a[n_] := Module[{p = 2, f = n!}, While[!PrimeQ[p * f + 1], p = NextPrime[p]]; p * f + 1]; Array[a, 20, 0] (* Amiram Eldar, Feb 25 2025 *)

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

a(n) = 1 + n!*A051888(n) = 1 + A000142(n)*A051888(n).

Offset changed to 0 and a(0) prepended by Amiram Eldar, Feb 25 2025

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

A051887 Minimal and special 2k-Germain primes, where 2k is in A002110 (primorial numbers).

Original entry on oeis.org

2, 2, 2, 2, 2, 5, 17, 11, 11, 11, 2, 23, 7, 43, 19, 3, 5, 2, 7, 3, 61, 53, 2, 41, 47, 2, 5, 7, 31, 2, 47, 13, 113, 7, 137, 103, 43, 41, 97, 3, 173, 97, 41, 13, 97, 59, 29, 53, 3, 107, 127, 197, 3, 487, 433, 31, 281, 587, 7, 89, 41, 47, 193, 239, 41, 7, 31, 67

Offset: 1

Labos Elemer, Dec 15 1999

nonn

a(n) is the minimal prime p such that primorial(n)*p + 1 is also prime.
While p is in A005384, the primorial(n)*p + 1 primes are in A051902 (primorial-safe primes).
Analogous to or subset of A051686, where the even numbers are 2, 6, ..., A002110(n), ...

			a(25) = 47 because primorial(25)*47 + 1 is also prime and minimal with this property: primorial(25)*47 + 1 = 47*2305567963945518424753102147331756070 + 1 = 108361694305439365963395800924592535291 is a minimal prime. The first 6 terms (2,2,2,2,2,5) correspond to first entries in A005384, A007693, A051645, A051647, A051653, A051654, respectively.

Amiram Eldar, Table of n, a(n) for n = 1..1000

Cf. A002110, A005384, A005385, A051686, A007693, A051886, A051888, A051902.

Table[p = 2; While[! PrimeQ[Product[Prime@ i, {i, n}] p + 1], p = NextPrime@ p]; p, {n, 68}] (* Michael De Vlieger, Jun 29 2017 *)

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

a(n) = (A051902(n)-1)/A002110(n). - Amiram Eldar, Feb 25 2025

More terms from Michael De Vlieger, Jun 29 2017

A064983 a(n) is the smallest prime p such that p*n! - 1 is prime.

Original entry on oeis.org

3, 3, 2, 2, 2, 2, 2, 2, 5, 3, 29, 11, 3, 5, 2, 2, 53, 2, 67, 79, 5, 61, 2, 7, 13, 5, 3, 11, 2, 107, 43, 7, 31, 199, 293, 17, 43, 197, 109, 41, 13, 277, 11, 167, 17, 83, 157, 31, 199, 131, 13, 5, 89, 47, 223, 83, 43, 7, 139, 151, 211, 19, 19, 23, 43, 311, 61, 53, 191, 163, 11

Offset: 0

Robert G. Wilson v, Oct 30 2001

nonn

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 (terms 0..200 from Harry J. Smith)

Cf. A051888.

Do[k = 1; While[ !PrimeQ[ Prime[k]*n! - 1], k++ ]; Print[ Prime[k]], {n, 1, 75} ]

{ allocatemem(932245000); for (n=0, 200, f=n!; k=1; while(!isprime(prime(k)*f - 1), k++); write("b064983.txt", n, " ", prime(k)) ) } \\ Harry J. Smith, Oct 02 2009

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).

A064998 a(n) is the smallest prime p such that p*n! +- 1 are twin primes.

Original entry on oeis.org

2, 2, 3, 2, 17, 7, 7, 3, 61, 29, 167, 401, 167, 19, 311, 461, 97, 919, 47, 71, 1483, 107, 13, 1571, 821, 1361, 769, 239, 163, 4229, 593, 373, 1571, 173, 6229, 3331, 1879, 2837, 2633, 12329, 2311, 269, 4159, 1217, 9719, 509, 3049, 8429, 307, 6121, 7919

Offset: 2

Robert G. Wilson v, Oct 31 2001

nonn

			a(6) = 17 because 17*6! = 12240, 12240 + 1 and 12240 - 1 are twin primes, and there is no prime less than 17 for which this pairing will work.

Harry J. Smith, Table of n, a(n) for n = 2..200

Cf. A051888, A064983.

Do[k = 1; While[ !PrimeQ[ Prime[k]*n! - 1] || !PrimeQ[ Prime[k]*n! + 1], k++ ]; Print[ Prime[k]], {n, 2, 75} ]

{ allocatemem(932245000); for (n=2, 200, f=n!; p=2; while (!isprime(p*f + 1) || !isprime(p*f - 1), p=nextprime(p + 1)); write("b064998.txt", n, " ", p) ) } \\ Harry J. Smith, Oct 03 2009

cp's OEIS Frontend

A051901 Minimal factorial safe-primes: a prime p = a(n) here if (p-1)/n! = A051888(n).

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

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A051887 Minimal and special 2k-Germain primes, where 2k is in A002110 (primorial numbers).

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A064983 a(n) is the smallest prime p such that p*n! - 1 is prime.

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

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A064998 a(n) is the smallest prime p such that p*n! +- 1 are twin primes.

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