A023201 -id:A023201 - cp's OEIS frontend

A049485 Primes p such that p + 510510 is also prime, where 510510 is the 7th primorial number A002110(7).

19, 41, 43, 59, 71, 73, 79, 101, 103, 107, 109, 167, 173, 181, 197, 199, 241, 257, 263, 283, 293, 307, 313, 317, 337, 379, 397, 409, 421, 431, 433, 479, 491, 503, 509, 523, 547, 577, 599, 601, 613, 641, 643, 653, 659, 661, 683, 691, 701, 727, 733, 751, 769

Offset: 1

Labos Elemer

nonn

p and p+510510 are not necessarily consecutive primes.

			19 is a term since it is prime and 19 + 510510 = 510529 is also prime.

Daniel Starodubtsev, Table of n, a(n) for n = 1..10000

Cf. A002110, A045320.

Cf. A001359, A023201, A049481, A049482, A049483, A049484, A154114.

lst={};Do[p=Prime[n];If[PrimeQ[p+510510],AppendTo[lst,p]],{n,6!}];lst (* Vladimir Joseph Stephan Orlovsky, Jan 04 2009 *)

isok(p) = isprime(p) && isprime(p + 510510); \\ Amiram Eldar, Mar 15 2025

A088766 a(n) = (A087681(n)-1)/2.

Original entry on oeis.org

5, 6, 8, 11, 12, 17, 18, 23, 26, 32, 33, 36, 38, 47, 51, 53, 66, 71, 72, 78, 86, 92, 93, 102, 108, 116, 117, 122, 128, 131, 137, 138, 143, 171, 176, 186, 197, 201, 207, 212, 213, 218, 227, 236, 242, 246, 248, 257, 281, 296, 303, 306, 312, 318, 323, 326, 333, 366

Offset: 1

Ray Chandler, Oct 26 2003

nonn

Numbers k such that 2*k + 1 - 6 and 2*k + 1 + 6 are sexy primes. [Jonathan Vos Post, Feb 14 2011]

			1002 is in the sequence because 2*1002 + 1 - 6 = 1999 is prime, and 2*1002 + 1 + 6 = 2011 is prime.

Vincenzo Librandi, Table of n, a(n) for n = 1..7600

Cf. A023201, A046117, A087681, A088762, A088764, A088765, A088768, A088770, A186243.

[n-1: n in [3..400] |IsPrime(2*n+5) and IsPrime(2*n-7)]; // Vincenzo Librandi, May 20 2017

Select[Range[3, 1000], PrimeQ[2 # + 5] && PrimeQ[2 # - 7] &] - 1 (* Vincenzo Librandi, May 20 2017 *)

{k such that 2*k + 1 - 6 is in A023201} = {k such that 2*k + 1 + 6 is in A046117}.

A140555 Primes p such that p + 6 is not a prime.

Original entry on oeis.org

2, 3, 19, 29, 43, 59, 71, 79, 89, 109, 113, 127, 137, 139, 149, 163, 179, 181, 197, 199, 211, 229, 239, 241, 269, 281, 283, 293, 313, 317, 337, 349, 359, 379, 389, 397, 401, 409, 419, 421, 431, 439, 449, 463, 467, 479, 487, 491, 499, 509, 521, 523, 547, 569

Offset: 1

Juri-Stepan Gerasimov, Jul 03 2008

nonn

Note that if Goldbach's Conjecture (2n = p1 + p2 for all n>=2) is false and K is the smallest value of n for which it fails, then for 2(K-3) = p3 + p4, the primes p3 and p4 must be taken from this list. See also A067775. - Keith Backman, Apr 05 2012

Zak Seidov, Table of n, a(n) for n = 1..10000

Cf. A000040, A023201, A067775.

Select[Prime[Range[200]],!PrimeQ[#+6]&] (* Harvey P. Dale, Dec 21 2016 *)

forprime(p=2,600,if(!isprime(p+6),print1(p,","))) \\ Klaus Brockhaus, Aug 12 2008

A000040 SET MINUS A023201. - R. J. Mathar, Aug 09 2008

Corrected by R. J. Mathar and Klaus Brockhaus, Aug 12 2008

A154114 Primes p such that p + 9699690 is also prime, where 9699690 is the 8th primorial number A002110(8).

Original entry on oeis.org

23, 37, 41, 43, 59, 73, 79, 83, 109, 113, 127, 137, 151, 163, 197, 199, 223, 227, 229, 233, 239, 251, 263, 269, 283, 313, 337, 349, 373, 383, 389, 409, 421, 449, 457, 463, 479, 523, 557, 599, 617, 647, 691, 727, 739, 743, 751, 757, 761, 773, 797, 811, 821

Offset: 1

Vladimir Joseph Stephan Orlovsky, Jan 04 2009

nonn

Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..2000 from G. C. Greubel)

Cf. A001359, A002110, A023201, A049481, A049482, A049483, A049484, A049485.

[p: p in PrimesUpTo(1000) | IsPrime(p+9699690)]; // Vincenzo Librandi, Sep 02 2016

lst={};Do[p=Prime[n];If[PrimeQ[p+9699690],AppendTo[lst,p]],{n,6!}];lst
Select[Prime[Range[200]],PrimeQ[#+9699690]&]  (* Harvey P. Dale, Apr 26 2011 *)

is(n)=isprime(n+9699690) && isprime(n) \\ Charles R Greathouse IV, Sep 02 2016

a(n) >> n log^2 n. - Charles R Greathouse IV, Sep 02 2016

A164574 Numbers k such that k and k+6 are both prime powers.

Original entry on oeis.org

1, 2, 3, 5, 7, 11, 13, 17, 19, 23, 25, 31, 37, 41, 43, 47, 53, 61, 67, 73, 83, 97, 101, 103, 107, 121, 125, 131, 151, 157, 163, 167, 173, 191, 193, 223, 227, 233, 251, 257, 263, 271, 277, 283, 307, 311, 331, 337, 343, 347, 353, 361, 367, 373, 383, 433, 443, 457

Offset: 1

Daniel Forgues, Aug 16 2009

nonn

Numbers n such that n + (0, 6) is a prime power pair.
n + (0, 2m), m >= 1, being an admissible pattern for prime pairs, since (0, 2m) = (0, 0) (mod 2), has high density.
n + (0, 2m-1), m >= 1, being a non-admissible pattern for prime pairs, since (0, 2m-1) = (0, 1) (mod 2), has low density [the only possible pairs are (2^a - 2m-1, 2^a) or (2^a, 2^a + 2m-1), a >= 0.]

Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..2492 from Daniel Forgues)

Cf. A023201, A000961.

k and (x) are prime powers: A006549 (k+1) A120431 (k+2), A164571 (k+3), A164572 (k+4), A164573 (k+5), this sequence (k+6).

Join[{1},Select[Range[500],AllTrue[{#,#+6},PrimePowerQ]&]] (* The program uses the AllTrue function from Mathematica version 10 *) (* Harvey P. Dale, Sep 30 2018 *)

is(n)=if(n<4,return(n>0)); isprimepower(n) && isprimepower(n+6) \\ Charles R Greathouse IV, Apr 24 2015

Edited by Daniel Forgues, Aug 17 2009

A230217 List of those primes p with p + 6 and 3*p + 8 also prime.

Original entry on oeis.org

5, 7, 11, 13, 17, 31, 41, 47, 53, 61, 73, 83, 101, 103, 131, 151, 157, 167, 193, 223, 251, 263, 271, 277, 307, 311, 347, 367, 433, 563, 571, 593, 601, 613, 641, 647, 677, 733, 823, 857, 977, 1097, 1117, 1217, 1223, 1231, 1291, 1301, 1361, 1427

Offset: 1

Zhi-Wei Sun, Oct 11 2013

nonn

Clearly, no term is congruent to 4 modulo 5.

This sequence is interesting because of the conjecture in the comments in A230219.

			a(1) = 5 since neither 2 + 6 nor 3 + 6 is prime, but 5 + 6 = 11 and 3*5 + 8 = 23 are both prime.

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

Cf. A023201, A230219.

PQ[p_]:=PrimeQ[p+6]&&PrimeQ[3p+8]
m=0
Do[If[PQ[Prime[n]],m=m+1;Print[m," ",Prime[n]]],{n,1,225}]
Select[Prime[Range[300]],AllTrue[{#+6,3#+8},PrimeQ]&] (* Harvey P. Dale, Sep 01 2023 *)

A049483 Primes p such that p + 2310 is also prime, where 2310 is the 5th primorial number A002110(5).

Original entry on oeis.org

23, 29, 31, 37, 41, 47, 61, 67, 71, 73, 79, 83, 89, 101, 107, 113, 127, 131, 137, 149, 157, 163, 167, 193, 211, 229, 233, 239, 241, 269, 281, 283, 307, 311, 337, 347, 349, 353, 367, 373, 379, 383, 389, 397, 401, 409, 419, 421, 431, 439, 443, 457, 467, 479

Offset: 1

Labos Elemer

nonn

p and p+2310 are not necessarily consecutive primes.

			23 is a term since it is prime and 23 + 2310 = 2333 is also prime.

Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..1000 from Harvey P. Dale)

Cf. A002110, A045320.

Cf. A001359, A023201, A049481, A049482, A049484, A049485, A154114.

Select[Prime[Range[100]],PrimeQ[#+2310]&] (* Harvey P. Dale, Nov 15 2012 *)

isok(p) = isprime(p) && isprime(p + 2310); \\ Amiram Eldar, Mar 15 2025

A049484 Primes p such that p + 30030 is also prime, where 30030 is the 6th primorial number A002110(6).

Original entry on oeis.org

17, 29, 41, 59, 61, 67, 73, 79, 83, 89, 103, 107, 109, 131, 139, 151, 157, 167, 173, 181, 193, 211, 223, 229, 239, 241, 263, 277, 283, 293, 311, 317, 337, 359, 373, 397, 401, 419, 439, 461, 463, 467, 479, 487, 499, 509, 523, 547, 563, 601, 607, 613, 619, 631

Offset: 1

Labos Elemer

nonn

p and p+30030 are not necessarily consecutive primes.

			17 is a term since it is prime and 17 + 30030 = 30047 is also prime.

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

Cf. A002110, A045320.

Cf. A001359, A023201, A049481, A049482, A049483, A049485, A154114.

Select[Prime[Range[150]],PrimeQ[#+30030]&] (* Harvey P. Dale, Sep 21 2022 *)

isok(p) = isprime(p) && isprime(p + 30030); \\ Amiram Eldar, Mar 15 2025

A103523 Concatenations of pairs of primes that differ by 100.

Original entry on oeis.org

3103, 7107, 13113, 31131, 37137, 67167, 73173, 79179, 97197, 127227, 139239, 151251, 157257, 163263, 181281, 193293, 211311, 283383, 331431, 349449, 367467, 379479, 409509, 421521, 457557, 463563, 487587, 499599, 541641, 547647, 577677

Offset: 1

Jonathan Vos Post, Mar 21 2005

Integers in this sequence can never be prime, as, starting from the second one, they are all multiples of 3.

			9191019 is in this sequence because 919 is prime, 919+100 = 1019 is prime and 9191019 is the concatenation of those two primes differing by 100.

Robert Israel, Table of n, a(n) for n = 1..9832

Cf. A000040, A001358, A023201, A100750, A103195, A103206, A104718, A104719.

f:= proc(n) if isprime(n) and isprime(n+100) then 10^(1+ilog10(n+100))*n+n+100 fi end proc:
map(f, [3,seq(i,i=7..1000,6)]); # Robert Israel, Dec 07 2015

FromDigits[Join@@IntegerDigits/@{#,#+100}]&/@Select[Prime@Range@200,PrimeQ[#+100]&] (* Giorgos Kalogeropoulos, Jul 04 2021 *)

from sympy import isprime, primerange as prange
def auptop(lim):
  return [int(str(p)+str(p+100)) for p in prange(2, lim+1) if isprime(p+100)]
print(auptop(577)) # Michael S. Branicky, Jul 04 2021

List: concatenate(p, p+100) iff p and p+100 are primes.

A104227 Primes one less than the sum over a sexy prime pair.

Original entry on oeis.org

19, 31, 67, 79, 127, 139, 151, 199, 211, 307, 547, 619, 739, 751, 919, 1087, 1231, 1459, 1471, 1759, 1987, 2131, 2179, 2239, 2251, 2467, 2647, 2851, 2971, 3319, 3331, 3391, 3499, 3511, 3559, 3571, 3727, 3739, 4027, 4567, 4759, 5107, 5347, 5419, 5431, 6367, 6607, 6619, 7027, 7219, 7459

Offset: 1

Giovanni Teofilatto, Apr 02 2005

Primes of the form A023201(i)+A046117(i)-1 or of the form 2*A087695(j)-1.

			19=7+13-1 is a prime and one less than the sum 7+13 over the second sexy prime pair.

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

Cf. A023201, A046117, A104228, A104229.

Select[2#+5&/@Select[Prime[Range[600]],PrimeQ[#+6]&],PrimeQ] (* Harvey P. Dale, Jan 04 2020 *)

Corrected definition. Extended beyond a(7). - R. J. Mathar, Nov 26 2008

cp's OEIS Frontend

A049485 Primes p such that p + 510510 is also prime, where 510510 is the 7th primorial number A002110(7).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

A088766 a(n) = (A087681(n)-1)/2.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Magma

Mathematica

Formula

A140555 Primes p such that p + 6 is not a prime.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

PARI

Formula

Extensions

A154114 Primes p such that p + 9699690 is also prime, where 9699690 is the 8th primorial number A002110(8).

Views

Author

Keywords

Links

Crossrefs

Programs

Magma

Mathematica

PARI

Formula

A164574 Numbers k such that k and k+6 are both prime powers.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

PARI

Extensions

A230217 List of those primes p with p + 6 and 3*p + 8 also prime.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

A049483 Primes p such that p + 2310 is also prime, where 2310 is the 5th primorial number A002110(5).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

A049484 Primes p such that p + 30030 is also prime, where 30030 is the 6th primorial number A002110(6).