A153218 -id:A153218 - cp's OEIS frontend

A023222 Primes p such that 6*p + 7 is also prime.

2, 5, 11, 17, 29, 31, 37, 61, 67, 71, 89, 101, 109, 127, 137, 167, 181, 191, 199, 229, 241, 257, 269, 277, 281, 311, 331, 337, 347, 359, 379, 389, 397, 419, 431, 457, 491, 499, 509, 541, 571, 577, 587, 601, 617, 631, 641, 647, 691, 709, 739, 751, 757, 769, 787, 797, 809

Offset: 1

David W. Wilson

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

Intersection of A153218 and A000040.

Cf. A153219.

[n: n in PrimesUpTo(900) | IsPrime(6*n+7)]; // Vincenzo Librandi, Nov 19 2010

Select[Prime[Range[200]],PrimeQ[6#+7]&] (* Harvey P. Dale, Nov 25 2012 *)

is(n)=isprime(6*n+7) && isprime(n) \\ Charles R Greathouse IV, Apr 18 2020

A153219 Numbers n such that 6*n + 7 is not prime.

Original entry on oeis.org

3, 7, 8, 13, 14, 18, 19, 21, 23, 27, 28, 30, 33, 35, 38, 40, 41, 42, 43, 47, 48, 49, 52, 53, 56, 58, 59, 63, 64, 66, 68, 70, 73, 74, 77, 78, 79, 81, 83, 84, 85, 87, 88, 91, 92, 93

Offset: 1

Vincenzo Librandi, Dec 21 2008

One less than the associated entry in A046954. - R. J. Mathar, Jan 05 2011

			Distribution of the terms in the following triangular array:
*;
*,3;
*,*,7;
*,*,*,*;
*,8,*,*,19;
*,*,14,*,*,27;
*,*, *,*,*, *,*;
*,13,*,*,30,*,*,47;
*,*,21,*,*,40,*,*,59;
*,*,*, *,*, *,*,*, *,*;
*,18,*,*,41,*,*,64,*,*,87;
*,*,28,*,*,53,*,*,78,*,*,103; etc.
where * marks the non-integer values of (2*h*K + k + h - 3)/3 with h >= k >= 1. - _Vincenzo Librandi_, Jan 15 2013

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

Cf. A153218.

[n: n in [0..110] | not IsPrime(6*n+7)]; // Vincenzo Librandi, Jan 12 2013

Select[Range[0, 200], !PrimeQ[6 # + 7] &] (* Vincenzo Librandi, Jan 12 2013 *)

A307562 Numbers k such that both 6k + 1 and 6k + 7 are prime.

Original entry on oeis.org

1, 2, 5, 6, 10, 11, 12, 16, 17, 25, 26, 32, 37, 45, 46, 51, 55, 61, 62, 72, 76, 90, 95, 100, 101, 102, 121, 122, 125, 137, 142, 146, 165, 172, 177, 181, 186, 187, 205, 215, 216, 220, 237, 241, 242, 247, 257, 270, 276, 277, 282, 290, 291, 292, 296, 297, 310, 311, 312, 331, 332, 335, 347, 355, 356, 380, 381, 390

Offset: 1

Sally Myers Moite, Apr 14 2019

nonn

There are 138 such numbers between 1 and 1000.
Prime pairs that differ by 6 are called "sexy" primes. Other prime pairs that differ by 6 are of the form 6n - 1 and 6n + 5.
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.

			a(3) = 5, so 6(5) + 1 = 31 and 6(5) + 7 = 37 are both prime.

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).

For the primes see A023201, A046117.
Similar sequences for twin primes are A002822, A067611, for "cousin" primes A056956, A186243.
Intersection of A024899 and A153218.
Cf. also A307561, A307563.

Select[Range[400], AllTrue[6 # + {1, 7}, PrimeQ] &] (* Michael De Vlieger, Apr 15 2019 *)

isok(n) = isprime(6*n+1) && isprime(6*n+7); \\ Michel Marcus, Apr 16 2019

A307563 Numbers k such that both 6k - 1 and 6k + 7 are prime.

Original entry on oeis.org

1, 2, 4, 5, 9, 10, 12, 15, 17, 22, 25, 29, 32, 39, 44, 45, 60, 65, 67, 72, 75, 80, 82, 94, 95, 99, 100, 109, 114, 117, 120, 124, 127, 137, 152, 155, 164, 169, 172, 177, 185, 194, 199, 204, 205, 214, 215, 220, 229, 240, 242, 247, 254, 260, 262, 267, 269, 270, 289, 304, 312, 330, 334, 347, 355, 359, 369, 374, 379, 389

Offset: 1

Sally Myers Moite, Apr 14 2019

nonn

There are 140 such numbers between 1 and 1000.
These numbers correspond to all the prime pairs which differ by 8 except 3 and 11.
Numbers in this sequence are those which are not 6cd - c - d - 1, 6cd + c - d, 6cd - c + d or 6cd + c + d - 1, that is, they are not (6c - 1)d - c - 1, (6c - 1)d + c, (6c + 1)d - c or (6c + 1)d + c - 1.

			a(4) = 5, so 6(5) - 1 = 29 and 6(5) + 7 = 37 are both prime.

Robert Israel, 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).

The primes are A023202, A092402, A031926.
Similar sequences for twin primes are A002822, A067611, for "cousin" primes A056956, A186243.
Intersection of A024898 and A153218.
Cf. also A307561, A307562.

select(t -> isprime(6*t-1) and isprime(6*t+7), [$1..500]); # Robert Israel, May 27 2019

isok(n) = isprime(6*n-1) && isprime(6*n+7); \\ Michel Marcus, Apr 16 2019

A023258 Primes that remain prime through 2 iterations of function f(x) = 6x + 7.

Original entry on oeis.org

5, 17, 29, 37, 89, 127, 199, 229, 269, 347, 359, 379, 397, 499, 769, 809, 929, 947, 977, 1049, 1087, 1129, 1277, 1279, 1367, 1409, 1439, 1489, 1499, 1607, 1609, 1787, 2017, 2039, 2089, 2399, 2539, 2707, 2819, 2837, 2957, 3089, 3109, 3217, 3229, 3389, 3499

Offset: 1

David W. Wilson

nonn

Primes p such that 6*p+7 and 36*p+49 are also primes. - Vincenzo Librandi, Aug 04 2010

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

Subsequence of A023222, A153218. - John Cerkan, Sep 14 2016

[n: n in [1..100000] | IsPrime(n) and IsPrime(6*n+7) and IsPrime(36*n+49)] // Vincenzo Librandi, Aug 04 2010

nrpQ[n_]:=AllTrue[Rest[NestList[6#+7&,n,2]],PrimeQ]; Select[Prime[ Range[ 500]],nrpQ] (* Harvey P. Dale, Oct 12 2020 *)

a(n) = 7 or 9 (mod 10) for n > 1. - John Cerkan, Sep 14 2016

A023289 Primes that remain prime through 3 iterations of function f(x) = 6x + 7.

Original entry on oeis.org

5, 37, 127, 347, 977, 2017, 3607, 5477, 9137, 10487, 13687, 14057, 14107, 19037, 19697, 19727, 20507, 22157, 23887, 24097, 25237, 25307, 26717, 26777, 27107, 29347, 30697, 33757, 33997, 34667, 34847, 35117, 35227, 37057, 40577, 40627, 41177, 41597

Offset: 1

David W. Wilson

nonn

Primes p such that 6*p+7, 36*p+49 and 216*p+301 are also primes. - Vincenzo Librandi, Aug 04 2010

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

Subsequence of A023222, A023258, and A153218.

[n: n in [1..150000] | IsPrime(n) and IsPrime(6*n+7) and IsPrime(36*n+49) and IsPrime(216*n+301)] // Vincenzo Librandi, Aug 04 2010

a(n) == 7 (mod 10), for n > 1. - John Cerkan, Sep 21 2016

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A023222 Primes p such that 6*p + 7 is also prime.

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A153219 Numbers n such that 6*n + 7 is not prime.

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A307562 Numbers k such that both 6*k + 1 and 6*k + 7 are prime.

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A307563 Numbers k such that both 6k - 1 and 6k + 7 are prime.

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A023258 Primes that remain prime through 2 iterations of function f(x) = 6x + 7.

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Magma

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A023289 Primes that remain prime through 3 iterations of function f(x) = 6x + 7.

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Magma

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A307562 Numbers k such that both 6k + 1 and 6k + 7 are prime.