A023256 -id:A023256 - cp's OEIS frontend

A007693 Primes p such that 6*p + 1 is also prime.

2, 3, 5, 7, 11, 13, 17, 23, 37, 47, 61, 73, 83, 101, 103, 107, 131, 137, 151, 173, 181, 233, 241, 257, 263, 271, 277, 283, 293, 311, 313, 331, 347, 367, 373, 397, 443, 461, 467, 503, 557, 577, 593, 601, 607, 641, 653, 661, 683, 727, 751, 761, 773, 787, 797, 853

Offset: 1

N. J. A. Sloane and Robert G. Wilson v

Joe Roberts, Lure of the Integers, Math. Assoc. America, 1992, p. 83.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..1000 from Harvey P. Dale)
Andrew Granville, Sophie Germain's theorem for prime pairs p, 6p+1, J. Number Theory 27 (1987), no. 1, 63-72.

Cf. A002476, A016921, A024899, A051644, A091178, A023256 (subset: 6*(6p+1)+1 also prime).

Prime terms of A024899.

[n: n in [0..1000] | IsPrime(n) and IsPrime(6*n+1)]; // Vincenzo Librandi, Nov 18 2010

Select[Prime@Range[150], PrimeQ[6# + 1] &] (* Ray Chandler, Mar 14 2007  *)

isok(k) = isprime(k) && isprime(6*k+1); \\ Amiram Eldar, Feb 24 2025

a(n) = (A051644(n)-1)/6.

Extended by Ray Chandler, Mar 14 2007

A023287 Primes that remain prime through 3 iterations of function f(x) = 6x + 1.

Original entry on oeis.org

61, 101, 1811, 3491, 4091, 5711, 5801, 6361, 7121, 10391, 10771, 11311, 13421, 15131, 17791, 18911, 19471, 20011, 24391, 25601, 25951, 30091, 35251, 41911, 45631, 47431, 55631, 58711, 62921, 67891, 70451, 70571, 72271, 74051, 74161, 75431, 80471, 86341

Offset: 1

David W. Wilson

nonn

Primes p such that s1=p, s2=6*s1+1, s3=6*s2+1 and s4=6*s3+1 are primes forming a special chain of four primes. A fifth term in such a chain cannot arise. See A085956, A086361, A086362.

Entries in chains are congruent to {1,7,3,9} mod 10.

			First chain is {61, 367, 2203, 13219};
319th chain is {1291391, 7748347, 46490083, 278940499}.

John Cerkan, Table of n, a(n) for n = 1..10000

Cf. A085956, A086361, A086362, A023330, A059766.

Subsequence of A007693, A023256, and A024899.

[n: n in [1..150000] | IsPrime(n) and IsPrime(6*n+1) and IsPrime(36*n+7) and IsPrime(216*n+43)] // Vincenzo Librandi, Aug 04 2010

k=0; m=6; Do[s=Prime[n]; s1=m*s+1; s2=m*s1+1; s3=m*s2+1; If[PrimeQ[s1]&&PrimeQ[s2]&&PrimeQ[s3], k=k+1; Print[{k, n, s, s1, s2, s3}]], {n, 1, 100000}] (* edited by Zak Seidov, Feb 08 2011 *)
thrQ[n_]:=AllTrue[Rest[NestList[6#+1&,n ,3]],PrimeQ]; Select[Prime[Range[9000]],thrQ] (* Harvey P. Dale, Mar 03 2024 *)

{p, 6p+1, 36p+7, 216p+43} are all primes, where p is prime.

Additional comments from Labos Elemer, Jul 23 2003

A263309 Numbers k such that p=6k+1 and q=6k+1 are primes.

Original entry on oeis.org

1, 2, 6, 10, 12, 17, 25, 30, 40, 45, 46, 47, 52, 55, 61, 62, 66, 96, 100, 101, 110, 121, 125, 131, 142, 151, 156, 172, 177, 186, 195, 200, 220, 221, 230, 237, 242, 255, 261, 282, 296, 305, 312, 331, 332, 356, 360, 367, 370, 380, 381, 382, 391, 425, 432, 446, 461, 465, 475, 495, 506, 510, 527, 530

Offset: 1

Zak Seidov, Oct 13 2015

nonn

Subsequence of A024899.

The subsequence of primes in this sequence is A023256.

Cf. A023256, A024899.

isA263309 := proc(n)
    if isprime(6*n+1) then
        if isprime(36*n+7) then
            true;
        else
            false;
        end if;
    else
        false;
    end if;
end proc:
for n from 1 to 100 do
    if isA263309(n) then
        printf("%d,",n);
    end if;
end do: # R. J. Mathar, Oct 17 2015

Select[Range[1000],PrimeQ[p=6*#+1]&& PrimeQ[q=6*p+1]&]

isok(n) = isprime(p=6*n+1) && isprime(6*p+1); \\ Michel Marcus, Oct 17 2015

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A007693 Primes p such that 6*p + 1 is also prime.

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A023287 Primes that remain prime through 3 iterations of function f(x) = 6x + 1.

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A263309 Numbers k such that p=6k+1 and q=6k+1 are primes.

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