A051648 -id:A051648 - cp's OEIS frontend

A051647 Primes p such that 210*p + 1 is also prime.

2, 3, 5, 7, 11, 13, 17, 23, 29, 47, 53, 59, 67, 73, 83, 89, 101, 137, 139, 157, 163, 179, 181, 191, 193, 223, 229, 251, 271, 277, 281, 313, 317, 347, 349, 353, 359, 401, 419, 421, 431, 433, 449, 457, 463, 479, 523, 577, 599, 601, 631, 653, 701, 709, 719, 727

Offset: 1

Labos Elemer

Analogous to A005384, Sophie Germain primes.

A002110(4)*p + 1 = 210*p + 1 and p are both primes.

			11 is in the sequence because 210*11 + 1 = 2311 is also prime.

Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..1000 from Vincenzo Librandi)

Cf. A005384, A005385, A007693, A051648.

[p: p in PrimesUpTo(900) | IsPrime(210*p+1)]; // Vincenzo Librandi, Apr 11 2013

Select[Prime[Range[200]],PrimeQ[210#+1]&]  (* Harvey P. Dale, Apr 25 2011 *)

isok(k) = isprime(k) && isprime(210*k+1); \\ Amiram Eldar, Feb 24 2025

a(n) = (A051648(n)-1)/210. - Amiram Eldar, Feb 24 2025

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

A051647 Primes p such that 210*p + 1 is also prime.

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