A023228 -id:A023228 - cp's OEIS frontend

A075706 Safe primes (A005385) (p and (p-1)/2 are primes) such that 8*p+1 (A023228) is also prime.

5, 11, 107, 179, 347, 479, 1187, 1307, 1367, 1487, 1619, 2027, 2207, 2447, 2999, 3119, 3467, 4007, 4079, 4139, 4799, 5087, 5807, 5927, 5939, 6827, 7079, 7247, 8699, 9587, 9839, 10607, 12107, 12539, 12659, 14207, 15299, 16139, 16187, 17027

Offset: 1

Jani Melik, Oct 02 2002

nonn

			11 is a prime, so is (11-1)/2=5 and also 8*11+1=89; 107 is a prime, (107-1)/2=53 and 8*107+1=857, ...

Harvey P. Dale, Table of n, a(n) for n = 1..1000

Cf. A005384, A005385, A059455, A023228.

ts_sg8_var_pras := proc(nmax) local i,tren,atek; tren := 0: for i from 1 to nmax do atek := numtheory[safeprime](i): if (atek > tren) then if (isprime(atek)='true' and isprime(6*atek+1)='true') then tren := atek: fi; fi; od; end: seq(ts_sg8_var_pras(i), i=1..3000);

Select[Prime[Range[2000]],AllTrue[{(#-1)/2,8#+1},PrimeQ]&] (* The program uses the AllTrue function from Mathematica version 10 *) (* Harvey P. Dale, Jan 31 2020 *)

forprime(p=3,20000,if(isprime((p-1)/2),if(isprime(8*p+1),print1(p","))))

More terms from Ralf Stephan, Mar 19 2003

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

A122095 Primes p for which 8*p+1 divides 2^p-1.

Original entry on oeis.org

11, 29, 179, 239, 431, 761, 857, 941, 1367, 1667, 1871, 1877, 2411, 2837, 3041, 3119, 3329, 3347, 3767, 4289, 5021, 5087, 5231, 5261, 5717, 5861, 6449, 6917, 6959, 7079, 7211, 7919, 8429, 8741, 8867, 9341, 9461, 9851, 10211, 10979, 12107, 12437, 12479

Offset: 1

J. Lowell, Oct 17 2006

nonn

The first 962 terms, all those with n<500000, are also in A023228. - R. J. Mathar, Oct 20 2006

All terms are in A023228, i.e., such that 8p+1 is prime, since a divisor of 8p+1 would also divide M(p)=A000225(p) and thus be of the form 2kp+1, but it is easily checked that 8p+1 cannot be a multiple of 2p+1 (nor of 4p+1 or 6p+1, of course). - M. F. Hasler, Mar 21 2011

			29 is in this sequence because 2^29-1 is divisible by 8 * 29 + 1 = 233.

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A000225, A002515, A188130.

isA122095 := proc(n) RETURN( isprime(n) and ( (2^n-1) mod (8*n+1)) = 0 ) ; end: n := 1 : for a from 2 to 500000 do if isA122095(a) then print(n,a) ; n := n+1 ; fi ; od ; # R. J. Mathar, Oct 20 2006

Select[Prime[Range[1500]],Divisible[2^#-1,8#+1]&] (* Harvey P. Dale, Dec 18 2012 *)
Select[Prime[Range[1500]],PowerMod[2,#,8#+1]==1&] (* Harvey P. Dale, May 28 2015 *)

forprime( p=1,1e4, Mod(2,p*8+1)^p-1 || print1(p, ", "))

More terms from R. J. Mathar, Oct 20 2006

A023347 Primes which remain prime through 5 iterations of function f(x) = 8x + 1.

Original entry on oeis.org

831167, 1154567, 2502767, 3019787, 3675197, 5056577, 6352487, 14519177, 26724377, 43003577, 47378927, 47695607, 56406197, 86332457, 86611757, 99568757, 121967987, 126435527, 127990997, 128149127, 128975057, 145281557, 155715407

Offset: 1

David W. Wilson

nonn

			First chain is {831167, 6649337, 53194697, 425557577, 3404460617, 27235684937};
If p is congruent to {1,3,7,9} mod 10, then consecutive iterates are congruent to {9,5,7,3}, {3,1,7,5}, {5,9,7,1} respectively; so only 10k+7 may remain prime through five iterations, as sequence demonstrates nicely. - _Labos Elemer_, Jul 23 2003

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

Cf. A085956, A086361, A086362, A023330, A059766, A023287, A000668, A076481, A086127.

Subsequence of A005123, A023228, A023260, A023291, and A023319.

k=0; m=8; Do[s=Prime[n]; s1=m*s+1; s2=m*s1+1; s3=m*s2+1; s4=m*s3+1; s5=m*s4+1; If[PrimeQ[s]&&PrimeQ[s1]&&PrimeQ[s2]&&PrimeQ[s3]&&PrimeQ[s4]&&PrimeQ[s5], k=k+1; Print[s]], {n, 1, 1000000}]
it5Q[n_]:=AllTrue[Rest[NestList[8#+1&,n,5]],PrimeQ]; Select[Prime[Range[ 9*10^6]],it5Q] (* The program uses the AllTrue function from Mathematica version 10 *) (* Harvey P. Dale, Sep 12 2014 *)

{p, 8p+1, 64p+9, 512p+73, 4096p+585, 32768p+4681} are all primes, where the initial p is prime.

a(n) == 197 (mod 210). - John Cerkan, Nov 04 2016

A075704 p and 12*p+1 are both primes.

Original entry on oeis.org

3, 5, 13, 19, 23, 29, 31, 59, 61, 71, 73, 83, 89, 101, 103, 139, 149, 191, 199, 223, 229, 233, 269, 271, 281, 293, 311, 379, 383, 401, 409, 433, 463, 479, 503, 523, 569, 601, 631, 643, 661, 691, 719, 751, 761, 773, 811, 829, 839, 863, 883, 929, 953, 1009, 1013

Offset: 0

Jani Melik, Oct 02 2002

nonn

			5 is a prime and 12*5+1=61 is also a prime. 13 and 12*13+1=157 are both primes...

Harvey P. Dale, Table of n, a(n) for n = 0..1000

Cf. A005384, A023212, A007693, A023228, A023237.

ts_m_sophie_germain_pras := proc(n); if (isprime(n)='true' and isprime(12*n+1)='true') then RETURN(n); fi; end: seq(ts_m_sophie_germain_pras(i), i=1..2030);

Select[Prime[Range[300]],PrimeQ[12#+1]&] (* Harvey P. Dale, Feb 06 2012 *)

A023291 Primes that remain prime through 3 iterations of function f(x) = 8x + 1.

Original entry on oeis.org

2, 1487, 2207, 2927, 8807, 11117, 16187, 17657, 26357, 44927, 45377, 48497, 91757, 110237, 117167, 122327, 125387, 126107, 145007, 170927, 174527, 190787, 193847, 203897, 230567, 244247, 246017, 270287, 280547, 283937, 347957, 362237, 364337

Offset: 1

David W. Wilson

nonn

Primes p such that 8*p+1, 64*p+9 and 512*p+73 are also primes. - Vincenzo Librandi, Aug 04 2010

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

Subsequence of A005123, A023228, and A023260.

[n: n in [1..450000] | IsPrime(n) and IsPrime(8*n+1) and IsPrime(64*n+9) and IsPrime(512*n+73)]; // Vincenzo Librandi, Aug 04 2010

okQ[n_]:=And@@PrimeQ[NestList[8#+1&,n,3]]; Select[Prime[Range[50000]],okQ] (* Harvey P. Dale, Jan 09 2011 *)

a(n) == 17 (mod 30) for n > 1. - John Cerkan, Sep 23 2016

A158014 Primes p such that (p-1)/8 is also prime.

Original entry on oeis.org

17, 41, 89, 137, 233, 569, 809, 857, 1049, 1097, 1193, 1433, 1913, 2153, 2777, 3209, 3449, 3593, 3833, 3929, 4073, 4457, 4793, 4937, 5273, 5417, 6089, 6473, 6569, 6857, 7433, 7529, 7577, 7817, 9209, 9497, 9833

Offset: 1

Roger L. Bagula, Mar 11 2009

Vincenzo Librandi, Table of n, a(n) for n = 1..1000

Cf. A005385 for (p-1)/2, A090866 for (p-1)/4, A051644 for (p-1)/6, A055781 for (p-1)/10.

Flatten[Table[If[PrimeQ[n] && PrimeQ[(n - 1)/8], n, {}], {n, 1, 10000}]]
Select[Prime[Range[1500]], PrimeQ[(# - 1) / 8]&] (* Vincenzo Librandi, Apr 14 2013 *)

list(lim)=my(v=List()); forprime(p=2,(lim-1)\8, if(isprime(8*p+1), listput(v,8*p+1))); Vec(v) \\ Charles R Greathouse IV, Oct 20 2021

a(n)=8*A023228(n)+1. - R. J. Mathar, Mar 15 2009

a(n) >> n log^2 n. - Charles R Greathouse IV, Oct 21 2021

Edited by the Associate Editors of the OEIS, Apr 22 2009

A228857 Odd primes p > 3 for which 14*p+1 is also prime.

Original entry on oeis.org

5, 17, 47, 53, 59, 83, 107, 113, 149, 167, 173, 239, 269, 353, 419, 443, 449, 503, 509, 563, 587, 599, 647, 659, 677, 719, 797, 827, 929, 947, 977, 983, 1097, 1103, 1109, 1187, 1193, 1223, 1229, 1259, 1289, 1367, 1409, 1427, 1433, 1439, 1493, 1523, 1667

Offset: 1

Ant King, Sep 06 2013

nonn

In 1823, Legendre proved that the first case of Fermat’s Last Theorem is true for all exponents that are members of this sequence (see Ribenboim’s reference, p.112).

			As both 5 and 14*5 + 1 = 71 are prime, then 5 is a member of this sequence.

Paulo Ribenboim; Fermat’s Last Theorem For Amateurs, Springer-Verlag, New York, Inc., (1999).

Cf. A005384, A023212, A023228, A023237, A155943.

[p: p in PrimesInInterval(5,2000) |IsPrime(14*p+1)]; // Vincenzo Librandi, Sep 18 2016

Select[Prime[Range[3,1667]],PrimeQ[14#+1] &]

lista(nn) = forprime(p=5, nn, if(isprime(14*p+1), print1(p, ", "))); \\ Altug Alkan, Sep 18 2016

cp's OEIS Frontend

A075706 Safe primes (A005385) (p and (p-1)/2 are primes) such that 8*p+1 (A023228) is also 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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A122095 Primes p for which 8*p+1 divides 2^p-1.

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Maple

Mathematica

PARI

Extensions

A023347 Primes which remain prime through 5 iterations of function f(x) = 8x + 1.

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Mathematica

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A075704 p and 12*p+1 are both primes.

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Maple

Mathematica

A023291 Primes that remain prime through 3 iterations of function f(x) = 8x + 1.

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Magma

Mathematica

Formula

A158014 Primes p such that (p-1)/8 is also prime.

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Programs

Mathematica

PARI

Formula