A051651 -id:A051651 - cp's OEIS frontend

A051654 Primes p such that 30030*p + 1 is also prime.

5, 11, 13, 29, 31, 41, 47, 61, 67, 79, 89, 127, 139, 149, 179, 181, 191, 197, 269, 307, 337, 347, 367, 383, 421, 467, 487, 541, 547, 569, 613, 641, 727, 809, 811, 821, 823, 883, 887, 941, 1031, 1033, 1087, 1093, 1103, 1117, 1193, 1201, 1237, 1259, 1279, 1283

Offset: 1

Labos Elemer

nonn

			p = 5 and 30030*p + 1 = 150151 are both primes.

Michael De Vlieger, Table of n, a(n) for n = 1..10000

Cf. A005384, A002110, A005385, A051651.

[p: p in PrimesUpTo (1300) | IsPrime(30030*p+1)]; // Vincenzo Librandi, Aug 16 2017

Select[Prime@ Range@ 210, PrimeQ[30030 # + 1] &] (* Michael De Vlieger, Aug 15 2017 *)

isok(p) = isprime(p) && isprime(30030*p+1); \\ Michel Marcus, Dec 19 2013

a(n) = (A051651(n)-1)/30030. - Amiram Eldar, Feb 24 2025

More terms from Michel Marcus, Dec 19 2013

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).

cp's OEIS Frontend

A051654 Primes p such that 30030*p + 1 is also prime.

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