A023204 -id:A023204 - cp's OEIS frontend

A190478 a(n) is the smallest prime prime(k) > a(n-1) such that the n numbers 2*prime(j)+3, j=k to k+n-1, are all prime.

2, 5, 13, 3767, 19913, 726109, 4827859, 59069473, 179993463679, 2280987436223

Offset: 1

Pierre CAMI, May 11 2011

This essentially searches for blocks of n consecutive primes of the form A023204 (see also A089530) with a minimum of the primes in the block set by the previous entry in the sequence. - R. J. Mathar, Jun 02 2011

Any further terms are > 10^13. - Lucas A. Brown, Mar 17 2024

			For n=1, 2 is prime and 2*2+3=7 is prime so a(1)=2.
For n=2, 5,7 are consecutive primes 2*5+3 and 2*7+3 are primes so a(2)=5 as 5 is the least such prime > 2.
For n=3, 13,17,19 are consecutive primes 2*13+3, 2*17+3, 2*19+3 are primes so a(3)=13 as 13 is the least such prime > 5.

Lucas A. Brown, Python program.

Cf. A023204.

isA023204 := proc(n) isprime(n) and isprime(2*n+3) ; end proc:
A190478idx := proc(n) option remember; if n = 1 then 1; else for a from procname(n-1)+1 do krun := true; for k from a to a+n-1 do if not isA023204(ithprime(k)) then krun := false; break; end if; end do: if krun then return a; end if; end do: end if; end proc:
A190478 := proc(n) ithprime( A190478idx(n)) ; end proc: # R. J. Mathar, Jun 02 2011

old(p,k)=while(k--,p=precprime(p-1));p;
n=1;k=0;forprime(p=2,4e9,if(isprime(p<<1+3),if(k++==n,print1(old(p,n)", ");k--;n++),k=0)) \\ Charles R Greathouse IV, May 11 2011

a(8) from Charles R Greathouse IV, May 11 2011

a(9)-a(10) from Lucas A. Brown, Mar 17 2024

A230225 Primes p such that 2p+1 and 2p+3 are not prime.

Original entry on oeis.org

31, 37, 59, 61, 71, 79, 101, 103, 107, 109, 149, 151, 163, 181, 211, 241, 257, 263, 271, 311, 313, 317, 331, 347, 367, 373, 389, 401, 421, 433, 449, 457, 461, 479, 499, 521, 541, 569, 571, 577, 587, 601, 619, 631, 661, 673, 677, 691, 701, 709, 727, 733, 751

Offset: 1

Vincenzo Librandi, Oct 12 2013

			31 is in the sequence because 2*31+1=63 and 2*31+3=65 are not prime.

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

Cf. A005384, A023204, A053176, A089531, A126107, A163769, A230039, A230117.

[p: p in PrimesUpTo(2500)|not IsPrime(2*p+1) and not IsPrime(2*p+3)];

Select[Range[10^3], PrimeQ[#]&&!PrimeQ[2 # + 1]&&!PrimeQ[2 # + 3]&]
Select[Prime[Range[200]],NoneTrue[2#+{1,3},PrimeQ]&] (* Harvey P. Dale, Sep 19 2021 *)

A243630 Primes p such that 2*p^3 - 3 is also prime.

Original entry on oeis.org

2, 7, 11, 13, 47, 101, 107, 151, 163, 167, 251, 257, 401, 443, 467, 521, 571, 641, 653, 673, 797, 907, 911, 971, 983, 997, 1013, 1151, 1153, 1181, 1187, 1223, 1231, 1277, 1291, 1303, 1361, 1433, 1481, 1511, 1597, 1637, 1723, 1741, 1811, 1951, 2027, 2081, 2083, 2141, 2287, 2311

Offset: 1

Vincenzo Librandi, Jun 08 2014

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

Cf. A063908, A023204, A153507, A174363, A243595.

[p: p in PrimesUpTo(2500) | IsPrime(2*p^3 - 3)];

Select[Prime[Range[2500]], PrimeQ[2 #^3 - 3] &]

A290839 a(n) = smallest prime p such that 2p + 2n - 1 is prime.

Original entry on oeis.org

2, 2, 2, 3, 2, 2, 3, 2, 2, 3, 2, 5, 3, 2, 2, 7, 3, 2, 3, 2, 2, 3, 2, 7, 3, 2, 5, 3, 2, 2, 7, 3, 2, 3, 2, 2, 13, 3, 2, 3, 2, 11, 3, 2, 5, 7, 3, 2, 3, 2, 2, 3, 2, 2, 3, 2, 13, 7, 11, 5, 19, 3, 2, 3, 2, 5, 3, 2, 2, 7, 5, 5, 3, 2, 2, 7, 3, 2, 13, 3, 2, 3, 2, 7, 3, 2

Offset: 0

XU Pingya, Aug 12 2017

Iain Fox, Table of n, a(n) for n = 0..20000

Cf. A005384, A023204, A023205, A023206, A023207, A171517, A020483, A290838.

Cf. A067076 (indices n at which a(n) = 2).

Table[j=0; found=False; While[!found, j++; found=PrimeQ[2Prime[j]+2n-1]]; Prime[j], {n, 85}]

a(n) = {my(p=2); while(!isprime(2*p+2*n-1), p = nextprime(p+1)); p;} \\ Michel Marcus, Aug 12 2017

a(-n) = A290838(n+1). - Iain Fox, Dec 14 2017

a(0) prepended by Iain Fox, Dec 14 2017

A349327 Primes p such that 2*p^2 is a term of A179993.

Original entry on oeis.org

2, 3, 7, 13, 43, 127, 211, 293, 743, 757, 797, 811, 1429, 1597, 1721, 2087, 2113, 2239, 2269, 2297, 2381, 2423, 2647, 3079, 3121, 3221, 3863, 4229, 4271, 4957, 5209, 5333, 5923, 6299, 6691, 7127, 7237, 7349, 7757, 7853, 8329, 8513, 8539, 8807, 9127, 9311, 9631, 9661

Offset: 1

Amiram Eldar, Nov 15 2021

nonn

The numbers of the form 2*p^2 where p is a term of this sequence are the only nonsquarefree terms of A179993.

Equivalently, primes p such that p^2 - 2 and 2*p^2 - 1 are also primes, or primes p such that p^2 - 2 is a term of A023204.

			2 is a term since 2*2^2 = 8 = 1*8 = 2*4 is a term of A179993: 8 - 1 = 7 and 4 - 2 = 2 are both primes.
3 is a term since 2*3^2 = 18 = 1*18 = 2*9 = 3*6 is a term of A179993: 18 - 1 = 17, 9 - 2 = 7 and 6 - 3 = 3 are all primes.

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

Cf. A013929, A023204, A179993.
Intersection of A062326 and A106483.
The prime terms of A225098.

q[n_] := AllTrue[{n, n^2 - 2, 2*n^2 - 1}, PrimeQ]; Select[Range[10000], q]

from itertools import islice
from sympy import isprime, nextprime
def A349327(): # generator of terms
    n = 2
    while True:
        if isprime(n**2-2) and isprime (2*n**2-1): yield n
        n = nextprime(n)
A349327_list = list(islice(A349327(),20)) # Chai Wah Wu, Nov 15 2021

A153145 Primes p such that 2*p + 19 is also prime.

Original entry on oeis.org

2, 5, 11, 17, 41, 47, 59, 89, 107, 131, 137, 149, 167, 191, 251, 269, 311, 317, 389, 401, 419, 431, 461, 467, 479, 521, 587, 599, 641, 677, 797, 809, 839, 857, 929, 941, 947, 977, 1031, 1061, 1097, 1109, 1181, 1187, 1229, 1301, 1307, 1319, 1361, 1367, 1409

Offset: 1

Vincenzo Librandi, Dec 19 2008

			For n=2, 2*n+19 = 23 is prime, so 2 is in the sequence.

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

Cf. A153143 (m and 2*m+19 are both prime), A005384 (Sophie Germain primes, m and 2*m+1 are both prime), A023204 (m and 2*m+3 are both prime), A023205 (m and 2*m+5 are both prime), A023206 (m and 2*m+7 are both prime), A023207 (m and 2*m+9 are both prime).

[p: p in PrimesUpTo(1500) | IsPrime(2*p+19)];

Select[Prime[Range[2000]],PrimeQ[2 # + 19] &] (* Vincenzo Librandi, Oct 20 2012 *)

Edited, corrected and extended by Klaus Brockhaus, Dec 22 2008

A171518 Primes p such that 3*p-+8 are primes.

Original entry on oeis.org

5, 7, 13, 17, 53, 73, 83, 113, 127, 157, 193, 223, 277, 347, 367, 433, 613, 647, 673, 743, 797, 907, 937, 1117, 1217, 1373, 1427, 1483, 1543, 1597, 1637, 1667, 1877, 1933, 2027, 2237, 2297, 2447, 2647, 2687, 2843, 3083, 3137, 3613, 3797, 4073, 4463, 4483

Offset: 1

Vladimir Joseph Stephan Orlovsky, Dec 10 2009

nonn

			5 is in the sequence since 3*5-8=7 and 3*5+8=23 are primes.

Vincenzo Librandi, Table of n, a(n) for n = 1..3000

Cf. A000040, A005382, A005384, A023204-A023207, A063908-A063912, A103803, A124098, A125272, A171517

Select[Prime[Range[7! ]],PrimeQ[3*#-8]&&PrimeQ[3*#+8]&]
Select[Prime[Range[700]],AllTrue[3#+{8,-8},PrimeQ]&] (* Harvey P. Dale, Feb 10 2025 *)

A238699 Primes p such that 2p + 3 and 4p + 3 are both prime.

Original entry on oeis.org

2, 5, 7, 17, 19, 47, 67, 89, 157, 227, 229, 307, 349, 439, 467, 487, 509, 599, 647, 797, 929, 1039, 1187, 1217, 1237, 1259, 1307, 1427, 1447, 1567, 1789, 2027, 2309, 2467, 2539, 2707, 2789, 2819, 3167, 3457, 3499, 3659, 3877, 3919, 4057, 4079, 4157, 4289, 4297

Offset: 1

Ilya Lopatin, Mar 03 2014, following a suggestion by Juri-Stepan Gerasimov

nonn

Intersection of A023204 and A023213.

Primes in A115334.

			89 is in the sequence because 2*89 + 3 = 181 and 4*89 + 3 = 359 are both prime.

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000

Cf. A023204, A023213; A115334.

[p: p in PrimesUpTo(4500) | IsPrime(2*p+3) and IsPrime(4*p+3)]; // Bruno Berselli, Mar 03 2014

Select[Prime[Range[600]],AllTrue[{2#+3,4#+3},PrimeQ]&] (* Harvey P. Dale, Dec 24 2023 *)

select(p->isprime(2*p+3)&&isprime(4*p+3), primes(1000)) \\ Charles R Greathouse IV, Mar 06 2014

Edited by Bruno Berselli, Mar 03 2014

A247010 Primes p such that (p-3)/2 and 2*p+3 are both prime.

Original entry on oeis.org

7, 13, 17, 29, 89, 97, 137, 197, 229, 277, 337, 349, 397, 557, 617, 797, 929, 937, 1117, 1217, 1237, 1777, 2129, 2309, 2437, 2477, 2617, 2749, 2857, 2909, 3049, 3109, 3137, 3329, 3389, 4057, 4229, 4289, 4409, 5237, 5297, 5417, 5557, 5717, 5857, 6689

Offset: 1

Vincenzo Librandi, Sep 09 2014

A023204 INTERSECT A089531. After 7, all terms are obviously in A002144.

Conjecture: the sequence is infinite.

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

Cf. A023204, A023242, A089531.

[p: p in PrimesUpTo(7000) | IsPrime((p-3)div 2) and IsPrime(2*p+3)];

Select[Prime[Range[900]], And@@PrimeQ/@{(# - 3)/2, 2 # + 3} &]

is(n)=isprime(n) && isprime(2*n+3) && isprime((n-3)\2) \\ Charles R Greathouse IV, Sep 09 2014

def t(i): return 2*i+3
[t(p) for p in primes(5000) if is_prime(t(p)) and is_prime(t(t(p)))] # Bruno Berselli, Sep 09 2014

a(n) = 2*A023242(n) + 3. [Bruno Berselli, Sep 09 2014]

A258153 Numbers of the form p^2 + q with p, q and 2*p + 3 all prime.

Original entry on oeis.org

6, 7, 9, 11, 15, 17, 21, 23, 27, 28, 30, 32, 33, 35, 36, 38, 41, 42, 44, 45, 47, 48, 51, 52, 54, 56, 57, 60, 62, 63, 65, 66, 68, 71, 72, 75, 77, 78, 80, 83, 84, 86, 87, 90, 92, 93, 96, 98, 101, 102, 104, 105, 107, 108, 110, 111, 113, 114, 116, 117, 120, 122, 126, 128, 131, 132, 134, 135, 138, 141

Offset: 1

Zhi-Wei Sun, May 22 2015

nonn

The conjecture in A258141 asserts that any six consecutive positive integers contain at least a term of the current sequence.

			a(1) = 6 since 6 = 2^2 + 2 with 2 and 2*2+3 = 7 both prime.
a(2) = 7 since 7 = 2^2 + 3 with 2, 3, 2*2+3 all prime.

Zhi-Wei Sun, Table of n, a(n) for n = 1..10000

Cf. A000040, A023204, A258139, A258140, A258141.

n=0;Do[Do[If[PrimeQ[2Prime[k]+3]&&PrimeQ[m-Prime[k]^2],n=n+1;Print[n," ",m];Goto[aa]],{k,1,PrimePi[Sqrt[m]]}];
Label[aa];Continue,{m,1,141}]
Module[{pp=40},Select[Union[#[[1]]^2+#[[2]]&/@Select[Tuples[ Prime[ Range[ pp]],2],PrimeQ[2#[[1]]+3]&]],#<=Prime[pp]-4&]] (* Harvey P. Dale, Jul 24 2021 *)

cp's OEIS Frontend

A190478 a(n) is the smallest prime prime(k) > a(n-1) such that the n numbers 2*prime(j)+3, j=k to k+n-1, are all prime.

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A230225 Primes p such that 2*p+1 and 2*p+3 are not prime.

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Magma

Mathematica

A243630 Primes p such that 2*p^3 - 3 is also prime.

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Magma

Mathematica

A290839 a(n) = smallest prime p such that 2p + 2n - 1 is prime.

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Mathematica

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A349327 Primes p such that 2*p^2 is a term of A179993.

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Mathematica

Python

A153145 Primes p such that 2*p + 19 is also prime.

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Magma

Mathematica

Extensions

A171518 Primes p such that 3*p-+8 are primes.

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Examples

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Programs

Mathematica

A238699 Primes p such that 2p + 3 and 4p + 3 are both prime.

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Examples

A230225 Primes p such that 2p+1 and 2p+3 are not prime.