A059325 -id:A059325

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A023221

Primes p such that 6*p + 5 is also prime.

+10
15

2, 3, 7, 11, 13, 17, 29, 31, 37, 41, 43, 59, 71, 73, 79, 83, 97, 107, 109, 113, 137, 139, 151, 157, 163, 181, 191, 193, 197, 227, 239, 241, 251, 263, 269, 277, 307, 311, 317, 337, 347, 349, 367, 373, 389, 401, 409, 421, 431, 443, 449, 479, 487, 499, 503, 541, 557, 577, 587

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,1

LINKS

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

MATHEMATICA

Select[Prime[Range[1000]], PrimeQ[6 # + 5] &] (* Vincenzo Librandi, May 20 2014 *)

PROG

(Magma) [n: n in PrimesUpTo(100) | IsPrime(6*n+5)]; // Vincenzo Librandi, Nov 20 2010

CROSSREFS

Subsequence of A059325.

KEYWORD

nonn,easy

AUTHOR

David W. Wilson

STATUS

approved

A056956

Numbers n such that 6n+1 and 6n+5 are both primes.

+10
12

1, 2, 3, 6, 7, 11, 13, 16, 17, 18, 21, 27, 32, 37, 38, 46, 51, 52, 58, 63, 66, 73, 76, 77, 81, 83, 102, 107, 112, 123, 126, 128, 137, 142, 143, 146, 147, 151, 156, 161, 168, 181, 182, 202, 213, 216, 217, 237, 238, 241, 247, 248, 258, 261, 263, 266, 268, 277, 282

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,2

COMMENTS

Note that if prime p>3 then p mod 6 = 1 or 5.

LINKS

M. F. Hasler, Table of n, a(n) for n = 1..5000

FORMULA

a(n) = (A023200(n+1)-1)/6 = (A046132(n+1)-5)/6 = A047847(n+1)/3

a(n) = floor(A087679(n+1)/6). - M. F. Hasler, Apr 05 2017

EXAMPLE

a(2)=2 since 6*2+1=13 and 6*2+5=17 are both prime.

MATHEMATICA

Select[Range[300], And @@ PrimeQ /@ ({1, 5} + 6#) &] (* Ray Chandler, Jun 29 2008 *)

PROG

(PARI) is(n)=isprime(n*6+1)&&isprime(n*6+5) \\ M. F. Hasler, Apr 05 2017

CROSSREFS

Cf. A002822, A024898, A024899, A059325.

KEYWORD

nonn

AUTHOR

Henry Bottomley, Jul 18 2000

EXTENSIONS

Edited by N. J. A. Sloane, Nov 07 2006

STATUS

approved

A059324

Numbers n such that 6n + 5 is composite.

+10
6

5, 10, 12, 15, 19, 20, 23, 25, 26, 30, 33, 34, 35, 36, 40, 45, 47, 49, 50, 53, 54, 55, 56, 60, 61, 62, 65, 67, 68, 70, 72, 75, 78, 80, 82, 85, 87, 88, 89, 90, 91, 95, 96, 100, 101, 103, 104, 105, 110, 111, 114, 115, 117, 118, 120, 121, 122, 124, 125, 127, 129, 130

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,1

COMMENTS

Conjecture: There exists no pair of primes (p, q > p^2) such that q - p^2 = 6*n - 4 (see A138479). - Philippe LALLOUET (philip.lallouet(AT)orange.fr), Mar 20 2008

LINKS

G. C. Greubel, Table of n, a(n) for n = 1..5000

FORMULA

a(n) = A046953(n-1) - 1.

EXAMPLE

a(3) = 12 because 6*12 + 5 = 77 is composite.

MATHEMATICA

Select[Range[200], !PrimeQ[6#+5]&] (* Harvey P. Dale, Mar 13 2011 *)

PROG

(PARI) isok(n) = ! isprime(6*n+5); \\ Michel Marcus, Jan 06 2017

CROSSREFS

Complement of A059325.

Cf. A138479.

KEYWORD

nonn,easy

AUTHOR

Anton Joha, Jan 26 2001

EXTENSIONS

More terms from Henry Bottomley, Jan 29 2001

STATUS

approved

A023288

Primes that remain prime through 3 iterations of function f(x) = 6x + 5.

+10
3

2, 11, 13, 31, 71, 83, 151, 163, 193, 197, 317, 347, 373, 503, 577, 811, 911, 919, 1049, 1051, 1201, 1423, 1721, 1907, 2089, 2243, 2543, 2719, 2963, 3529, 3583, 3607, 3797, 4091, 4153, 4217, 4243, 4409, 4591, 4637, 4783, 5209, 5557, 5783, 5849, 5923, 6091

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,1

COMMENTS

Primes p such that 6*p+5, 36*p+35 and 216*p+215 are also primes. - Vincenzo Librandi, Aug 04 2010

LINKS

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

PROG

(Magma) [n: n in [1..150000] | IsPrime(n) and IsPrime(6*n+5) and IsPrime(36*n+35) and IsPrime(216*n+215)] // Vincenzo Librandi, Aug 04 2010

CROSSREFS

Subsequence of A023221, A023257, and A059325.

KEYWORD

nonn

AUTHOR

David W. Wilson

STATUS

approved

A307561

Numbers k such that both 6*k - 1 and 6*k + 5 are prime.

+10
3

1, 2, 3, 4, 7, 8, 9, 14, 17, 18, 22, 28, 29, 32, 38, 39, 42, 43, 44, 52, 58, 59, 64, 74, 77, 84, 93, 94, 98, 99, 107, 108, 109, 113, 137, 143, 147, 157, 158, 162, 163, 169, 182, 183, 184, 197, 198, 203, 204, 213, 214, 217, 227, 228, 238, 239, 247, 248, 249, 259, 267, 268, 269, 312, 317, 318, 329, 333, 344

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,2

COMMENTS

There are 146 terms below 10^3, 831 terms below 10^4, 5345 terms below 10^5, 37788 terms below 10^6 and 280140 terms below 10^7.

Prime pairs differing by 6 are called "sexy" primes. Other prime pairs with difference 6 are of the form 6n + 1 and 6n + 7.

Numbers in this sequence are those which are not 6cd + c - d - 1, 6cd + c - d, 6cd - c + d - 1 or 6cd - c + d, that is, they are not (6c - 1)d + c - 1, (6c - 1)d + c, (6c + 1)d - c - 1 or (6c + 1)d - c.

LINKS

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

Sally M. Moite, "Primeless" Sieves for Primes and for Prime Pairs Which Differ by 2m, vixra:1903.0353 (2019).

EXAMPLE

a(2) = 2, so 6(2) - 1 = 11 and 6(2) + 5 = 17 are both prime.

MATHEMATICA

Select[Range[500], PrimeQ[6# - 1] && PrimeQ[6# + 5] &] (* Alonso del Arte, Apr 14 2019 *)

PROG

(PARI) is(k) = isprime(6*k-1) && isprime(6*k+5); \\ Jinyuan Wang, Apr 20 2019

CROSSREFS

Primes differing from each other by 6 are A023201, A046117.

Similar sequences for twin primes are A002822, A067611, for "cousin" primes A056956, A186243.

Intersection of A024898 and A059325.

Cf. also A307562, A307563.

KEYWORD

nonn

AUTHOR

Sally Myers Moite, Apr 14 2019

STATUS

approved

A023317

Primes that remain prime through 4 iterations of function f(x) = 6x + 5.

+10
2

11, 13, 83, 151, 317, 373, 1721, 3529, 4153, 4243, 4637, 4783, 5209, 5849, 5923, 6661, 8431, 10903, 11329, 14519, 16183, 16979, 20149, 26669, 27509, 27827, 29873, 29947, 32987, 33637, 33937, 34919, 35099, 35543, 36277, 36691, 38069, 38461, 41651, 47407

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,1

COMMENTS

Primes p such that 6*p+5, 36*p+35, 216*p+215 and 1296*p+1295 are also primes. - Vincenzo Librandi, Aug 04 2010

LINKS

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

MATHEMATICA

if4Q[n_]:=AllTrue[Rest[NestList[6#+5&, n, 4]], PrimeQ]; Select[Prime[ Range[ 5000]], if4Q] (* The program uses the AllTrue function from Mathematica version 10 *) (* Harvey P. Dale, May 10 2018 *)

PROG

(Magma) [n: n in [1..1000000] | IsPrime(n) and IsPrime(6*n+5) and IsPrime(36*n+35) and IsPrime(216*n+215) and IsPrime(1296*n+1295)] // Vincenzo Librandi, Aug 04 2010

CROSSREFS

Subsequence of A023221, A023257, A023288, and A059325.

KEYWORD

nonn

AUTHOR

David W. Wilson

STATUS

approved

A173178

Numbers k such that 2*k+3 is a prime of the form 3*A024893(m) + 2.

+10
2

1, 4, 7, 10, 13, 19, 22, 25, 28, 34, 40, 43, 49, 52, 55, 64, 67, 73, 82, 85, 88, 94, 97, 112, 115, 118, 124, 127, 130, 133, 139, 145, 154, 157, 172, 175, 178, 190, 193, 199, 208, 214, 220, 223, 229, 232, 238, 244, 250, 253, 259, 277, 280, 283, 292, 295, 298, 307, 319

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,2

COMMENTS

With the Bachet-Bézout theorem implicating Gauss Lemma and the Fundamental Theorem of Arithmetic,

for k > 1, k = 2*a + 3*b (a and b integers)

first type

A001477 = (2*A080425) + (3*A008611)

A000040 = (2*A039701) + (3*A157966)

A024893 Numbers k such that 3*k + 2 is prime

A034936 Numbers k such that 3*k + 4 is prime

OR second type

A001477 = (2*A028242) + (3*A059841)

A000040 = (2*A067076) + (3*1)

A067076 Numbers k such that 2*k + 3 is prime

k a b OR a b

-- - - - -

0 0 0 0 0

1 - - - -

2 1 0 1 0

3 0 1 0 1

4 2 0 2 0

5 1 1 1 1

6 0 2 3 0

7 2 1 2 1

8 1 2 4 0

9 0 3 3 1

10 2 2 5 0

11 1 3 4 1

12 0 4 6 0

13 2 3 5 1

14 1 4 7 0

15 0 5 6 1

...

2* 1 + 3 OR 3* 1 + 2 = 5;

2* 4 + 3 OR 3* 3 + 2 = 11;

2* 7 + 3 OR 3* 5 + 2 = 17;

2*10 + 3 OR 3* 7 + 2 = 23;

2*13 + 3 OR 3* 9 + 2 = 29;

2*19 + 3 OR 3*13 + 2 = 41;

2*22 + 3 OR 3*15 + 2 = 47;

2*25 + 3 OR 3*17 + 2 = 53;

2*28 + 3 OR 3*19 + 2 = 59.

A024893 Numbers k such that 3k+2 is prime.

A007528 Primes of the form 6k-1.

A024898 Positive integers k such that 6k-1 is prime.

1, 4, 7, 10, 13, 19, ... = (3*(4*A024898 - A024893) - 7)/2 = (A112774 - 3*A024893 - 5)/2 = A003627 - (3*A024893 - 5)/2.

LINKS

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

Chris K. Caldwell, FAQ: Are all primes (past 2 and 3) of the forms 6n+1 and 6n-1?, Frequently asked questions about primes.

FORMULA

a(n) = 3*A059325(n) + 1. - Amiram Eldar, Jul 30 2024

MATHEMATICA

Select[Range[0, 320], PrimeQ[(p = 2*# + 3)] && Mod[p, 3] == 2 &] (* Amiram Eldar, Jul 30 2024 *)

CROSSREFS

Cf. A067076, A024893, A007528, A024898, A059325.

KEYWORD

nonn,easy,uned

AUTHOR

Eric Desbiaux, Feb 11 2010

EXTENSIONS

Data corrected and extended by Amiram Eldar, Jul 30 2024

STATUS

approved

A023345

Primes that remain prime through 5 iterations of function f(x) = 6x + 5.

+10
1

13, 4637, 5849, 5923, 16183, 16979, 34919, 36277, 67003, 79337, 115571, 159739, 175141, 245753, 249133, 305717, 341569, 359353, 383833, 437263, 455317, 498497, 511519, 567121, 579961, 581699, 633797, 683831, 693431, 849197, 972197, 1022449

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,1

COMMENTS

Primes p such that 6*p+5, 36*p+35, 216*p+215, 1296*p+1295 and 7776*p+7775 are also primes. - Vincenzo Librandi, Aug 05 2010

LINKS

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

MATHEMATICA

Select[Range[1100000], And@@PrimeQ[NestList[6#+5&, #, 5]]&] (* Harvey P. Dale, Mar 31 2012 *)

PROG

(Magma) [n: n in [1..10000000] | IsPrime(n) and IsPrime(6*n+5) and IsPrime(36*n+35) and IsPrime(216*n+215) and IsPrime(1296*n+1295) and IsPrime(7776*n+7775)] // Vincenzo Librandi, Aug 05 2010

CROSSREFS

Subsequence of A023221, A023257, A023288, A023317, and A059325.

KEYWORD

nonn

AUTHOR

David W. Wilson

STATUS

approved

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