A023210 -id:A023210 - cp's OEIS frontend

A023248 Primes that remain prime through 2 iterations of function f(x) = 3x + 8.

3, 11, 13, 31, 41, 43, 53, 101, 113, 211, 223, 263, 283, 431, 433, 491, 521, 563, 571, 601, 631, 641, 673, 743, 811, 911, 1151, 1361, 1621, 1693, 1973, 2243, 2393, 3083, 3163, 3181, 3343, 3461, 3821, 3923, 4481, 4523, 4561, 4591, 4663, 4861, 4903, 5051, 5261

Offset: 1

David W. Wilson

nonn

Primes p such that 3*p+8 and 9*p+32 are also primes. - Vincenzo Librandi, Aug 04 2010

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

Subsequence of A023210.

[n: n in [0..100000] | IsPrime(n) and IsPrime(3*n+8) and IsPrime(9*n+32)]; // Vincenzo Librandi, Aug 04 2010

Select[Range@ 5280, Times @@ Boole@ PrimeQ@ NestList[3 # + 8 &, #, 2] > 0 &] (* Michael De Vlieger, Sep 13 2016 *)
fQ[p_]:=AllTrue[Rest[NestList[3#+8&,p,2]],PrimeQ]; Select[Prime[Range[700]],fQ] (* Harvey P. Dale, Sep 01 2024 *)

a(n) == 1 or 3 (mod 10) for n >= 2. - John Cerkan, Sep 13 2016

A023279 Primes that remain prime through 3 iterations of function f(x) = 3x + 8.

Original entry on oeis.org

11, 31, 211, 1151, 3181, 5051, 5261, 6101, 6661, 9391, 9551, 10501, 11701, 13171, 15901, 16481, 19531, 22051, 24181, 26261, 27031, 28351, 28591, 28661, 29411, 30941, 31321, 32621, 38011, 40471, 42101, 48371, 49921, 57571, 59791, 61981, 66161, 67271

Offset: 1

David W. Wilson

nonn

Primes p such that 3*p+8, 9*p+32 and 27*p+104 are also primes. - Vincenzo Librandi, Aug 04 2010

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

Subsequence of A023210 and of A023248.

[n: n in [1..150000] | IsPrime(n) and IsPrime(3*n+8) and IsPrime(9*n+32) and IsPrime(27*n+104)] // Vincenzo Librandi, Aug 04 2010

Select[Prime@ Range@ 7000, Times @@ Boole@ PrimeQ@ Rest@ NestList[3 # + 8 &, #, 3] > 0 &] (* Michael De Vlieger, Sep 19 2016 *)

is(n)=isprime(n) && isprime(3*n+8) && isprime(9*n+32) && isprime(27*n+104) \\ Charles R Greathouse IV, Sep 20 2016

a(n) == 1 (mod 10). - John Cerkan, Sep 16 2016

A023309 Primes that remain prime through 4 iterations of function f(x) = 3x + 8.

Original entry on oeis.org

3181, 9551, 22051, 30941, 32621, 61981, 76651, 99961, 134291, 151901, 163661, 185951, 226691, 227671, 240551, 288191, 342821, 374501, 394411, 402881, 423781, 426301, 446461, 456151, 459091, 460951, 554011, 572321, 577601, 617191, 653831

Offset: 1

David W. Wilson

nonn

Primes p such that 3*p+8, 9*p+32, 27*p+104 and 81*p+320 are also primes. - Vincenzo Librandi, Aug 04 2010

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

Subsequence of A023210, A023248, and A023279.

[n: n in [1..1000000] | IsPrime(n) and IsPrime(3*n+8) and IsPrime(9*n+32) and IsPrime(27*n+104) and IsPrime(81*n+320)]; // Vincenzo Librandi, Aug 04 2010

okQ[n_]:=And@@PrimeQ/@Rest[NestList[3#+8&,n,4]]; Select[Prime[Range[60000]],okQ] (* Harvey P. Dale, Aug 16 2010 *)

a(n) == 1 or 31 (mod 70). - John Cerkan, Oct 04 2016

A023337 Primes that remain prime through 5 iterations of function f(x) = 3x + 8.

Original entry on oeis.org

3181, 61981, 134291, 342821, 459091, 882451, 984931, 1016011, 1028471, 1181701, 1391561, 1897801, 2009311, 2272301, 2476421, 2769791, 3048041, 3085421, 3128821, 3207221, 3545111, 4092931, 4690591, 5015321, 5863651, 6027941, 6361351, 6796541

Offset: 1

David W. Wilson

nonn

Primes p such that 3*p+8, 9*p+32, 27*p+104, 81*p+320, and 243*p+968 are also primes. - Vincenzo Librandi, Aug 05 2010

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

Subsequence of A023210, A023248, A023279, and A023309.

[n: n in [1..25000000] | IsPrime(n) and IsPrime(3*n+8) and IsPrime(9*n+32) and IsPrime(27*n+104) and IsPrime(81*n+320) and IsPrime(243*n+968)] // Vincenzo Librandi, Aug 05 2010

Select[Prime[Range[500000]],And@@PrimeQ[Rest[NestList[3#+8&,#,5]]]&] (* Harvey P. Dale, Apr 07 2014 *)

is(n)=isprime(n) && isprime(3*n+8) && isprime(9*n+32) && isprime(27*n+104) && isprime(81*n+320) && isprime(243*n+968) \\ Charles R Greathouse IV, Oct 11 2016

a(n) == 31 (mod 70). - John Cerkan, Oct 11 2016

A230243 Number of primes p < n with 3p + 8 and (p-1)n + 1 both prime.

Original entry on oeis.org

0, 0, 0, 0, 1, 1, 1, 1, 2, 2, 2, 1, 3, 1, 4, 2, 1, 4, 2, 2, 4, 2, 3, 2, 4, 3, 4, 4, 2, 2, 2, 1, 5, 3, 4, 3, 3, 2, 3, 4, 2, 2, 4, 2, 4, 4, 1, 5, 3, 2, 6, 4, 1, 5, 6, 3, 3, 5, 1, 5, 5, 2, 7, 5, 3, 4, 4, 3, 4, 6, 3, 4, 6, 4, 5, 6, 3, 7, 4, 2, 6, 1, 3, 5, 9, 3, 3, 7, 4, 3, 7, 1, 6, 5, 5, 5, 6, 3, 6, 7

Offset: 1

Zhi-Wei Sun, Oct 13 2013

nonn

Conjecture: a(n) > 0 for all n > 4.
This implies A. Murthy's conjecture (cf. A034693) that for any integer n > 1, there is a positive integer k < n such that k*n + 1 is prime.
Conjecture verified for n up to 10^9. - Mauro Fiorentini, Sep 21 2023

			a(8) = 1 since 8 = 3 + 5 with 3, 3*3+8 = 17, (3-1)*8+1 = 17 all prime.
a(17) = 1 since 17 = 7 + 10, and 7, 3*7+8 = 29, (7-1)*17+1 = 103 are all prime.

Zhi-Wei Sun, Table of n, a(n) for n = 1..10000
Zhi-Wei Sun, Conjectures involving primes and quadratic forms, arXiv:1211.1588 [math.NT], 2012-2017.

Cf. A034693, A085053, A086685, A023210, A219864, A230217, A230219, A230241.

a[n_]:=Sum[If[PrimeQ[3Prime[i]+8]&&PrimeQ[(Prime[i]-1)n+1],1,0],{i,1,PrimePi[n-1]}]
Table[a[n],{n,1,100}]

A352170 Primes p such that p+4, 3p+4 and 3p+8 are also prime.

Original entry on oeis.org

3, 13, 103, 223, 823, 2953, 7873, 11113, 11863, 13033, 13963, 16063, 22153, 23743, 24763, 27733, 30133, 31513, 34213, 35593, 39883, 41893, 43063, 50383, 51043, 54493, 62983, 65323, 66343, 68473, 71593, 72643, 87793, 88423, 98893, 101203, 106363, 110563, 127873, 134593, 136603, 158563, 164623, 165703

Offset: 1

J. M. Bergot and Robert Israel, Mar 07 2022

nonn

Members p of A023200 such that 3*p+4 is also in A023200.

Except for 3, all terms == 13 (mod 30).

			a(4) = 223 is a term because 223, 223+4 = 227, 3*223+4 = 673 and 3*223+8 = 677 are all prime.

Martin Ehrenstein, Table of n, a(n) for n = 1..10000

Intersection of A023200, A023209 and A023210.

select(p -> isprime(p) and isprime(p+4) and isprime(3*p+4) and isprime(3*p+8), [3,seq(i,i=13..10^6,30)]);

Select[Range[200000], AllTrue[{#, # + 4, 3*# + 4, 3*# + 8}, PrimeQ] &] (* Amiram Eldar, Mar 07 2022 *)

from sympy import sieve, isprime
for p in sieve.primerange(0, 10**6):
    if(all(isprime(q) for q in [p+4, 3*p+4, 3*p+8])):
        print (p, end=", ") # Martin Ehrenstein, Mar 09 2022

cp's OEIS Frontend

A023248 Primes that remain prime through 2 iterations of function f(x) = 3x + 8.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Magma

Mathematica

Formula

A023279 Primes that remain prime through 3 iterations of function f(x) = 3x + 8.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Magma

Mathematica

PARI

Formula

A023309 Primes that remain prime through 4 iterations of function f(x) = 3x + 8.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Magma

Mathematica

Formula

A023337 Primes that remain prime through 5 iterations of function f(x) = 3x + 8.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Magma

Mathematica

PARI

Formula

A230243 Number of primes p < n with 3*p + 8 and (p-1)*n + 1 both prime.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

A352170 Primes p such that p+4, 3*p+4 and 3*p+8 are also prime.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Mathematica

Python

A230243 Number of primes p < n with 3p + 8 and (p-1)n + 1 both prime.

A352170 Primes p such that p+4, 3p+4 and 3p+8 are also prime.