cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-7 of 7 results.

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

Views

Author

Ray Chandler, Oct 26 2003

Keywords

Comments

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

Examples

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

Links

Crossrefs

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

Programs

  • Magma
    [n-1: n in [3..400] |IsPrime(2*n+5) and IsPrime(2*n-7)]; // Vincenzo Librandi, May 20 2017
  • Mathematica
    Select[Range[3, 1000], PrimeQ[2 # + 5] && PrimeQ[2 # - 7] &] - 1 (* Vincenzo Librandi, May 20 2017 *)

Formula

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

A307561 Numbers k such that both 6*k - 1 and 6*k + 5 are prime.

Original entry on oeis.org

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
Offset: 1

Views

Author

Sally Myers Moite, Apr 14 2019

Keywords

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.

Examples

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

Links

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.

Programs

  • Mathematica
    Select[Range[500], PrimeQ[6# - 1] && PrimeQ[6# + 5] &] (* Alonso del Arte, Apr 14 2019 *)
  • PARI
    is(k) = isprime(6*k-1) && isprime(6*k+5); \\ Jinyuan Wang, Apr 20 2019

A307562 Numbers k such that both 6*k + 1 and 6*k + 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

Views

Author

Sally Myers Moite, Apr 14 2019

Keywords

Comments

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.

Examples

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

Links

Crossrefs

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.

Programs

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

Views

Author

Sally Myers Moite, Apr 14 2019

Keywords

Comments

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.

Examples

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

Links

Crossrefs

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.

Programs

  • Maple
    select(t -> isprime(6*t-1) and isprime(6*t+7), [$1..500]); # Robert Israel, May 27 2019
  • PARI
    isok(n) = isprime(6*n-1) && isprime(6*n+7); \\ Michel Marcus, Apr 16 2019

A199717 Numbers k such that 6*k-1 is composite, but 6*k-5 is prime.

Original entry on oeis.org

6, 11, 13, 24, 26, 27, 31, 34, 36, 41, 46, 48, 56, 57, 62, 63, 69, 71, 73, 88, 91, 92, 96, 97, 101, 102, 104, 106, 111, 116, 119, 122, 123, 126, 132, 136, 139, 154, 166, 167, 171, 173, 174, 176, 178, 179, 187, 188, 189, 193, 196, 201, 206, 207, 209, 216, 221
Offset: 1

Views

Author

Eleonora Echeverri-Toro, Nov 09 2011

Keywords

Links

Crossrefs

Cf. A186243.

Programs

  • Magma
    [ p div 6 +1: p in PrimesUpTo(1326) | not IsPrime(p+4) and p mod 6 eq 1 ]; // Bruno Berselli, Nov 09 2011
  • Mathematica
    Select[Range[221], PrimeQ[6 # - 5] && ! PrimeQ[6 # - 1] &] (* T. D. Noe, Nov 09 2011 *)
  • PARI
    is(n)=isprime(6*n-5) && !isprime(6*n-1) \\ Charles R Greathouse IV, Jun 13 2017

A199718 Numbers k such that 6*k-5 is composite, but 6*k-1 is prime.

Original entry on oeis.org

5, 9, 10, 15, 23, 25, 29, 30, 32, 40, 42, 43, 44, 45, 49, 58, 60, 65, 70, 72, 75, 80, 85, 87, 93, 94, 95, 98, 99, 100, 107, 109, 110, 114, 117, 120, 133, 135, 137, 140, 155, 158, 159, 163, 164, 170, 172, 175, 177, 184, 185, 192, 194, 197, 198, 199, 204, 205
Offset: 1

Views

Author

Eleonora Echeverri-Toro, Nov 09 2011

Keywords

Links

Crossrefs

Cf. A186243.

Programs

  • Magma
    [ p div 6 +1: n in [4..204] | not IsPrime(p-4) and p mod 6 eq 5 where p is NthPrime(n) ]; // Bruno Berselli, Nov 09 2011
  • Mathematica
    Select[Range[2, 205], ! PrimeQ[6 # - 5] && PrimeQ[6 # - 1] &] (* T. N. Noe, Nov 09 2011 *)

A199716 Numbers k such that 6k-5 and 6k-1 are both composite.

Original entry on oeis.org

16, 20, 21, 35, 37, 50, 51, 54, 55, 61, 66, 68, 76, 79, 81, 83, 86, 89, 90, 105, 112, 115, 118, 121, 125, 128, 130, 131, 134, 141, 142, 145, 146, 149, 150, 151, 153, 156, 160, 161, 165, 168, 180, 181, 186, 190, 191, 195, 200, 202, 208, 211, 212, 219, 223
Offset: 1

Views

Author

Eleonora Echeverri-Toro, Nov 09 2011

Keywords

Crossrefs

Cf. A186243.

Programs

  • Magma
    [ n: n in [1..223] | not IsPrime(6*n-5) and not IsPrime(6*n-1) ]; // Bruno Berselli, Nov 09 2011
  • Mathematica
    Select[Range[223], ! PrimeQ[6#-5] && ! PrimeQ[6#-1] &] (* T. D. Noe, Nov 09 2011 *)
Select[Range[250],AllTrue[6#+{-5,-1},CompositeQ]&] (* Requires Mathematica version 10 or later *)  (* Harvey P. Dale, Oct 11 2018 *)
  • PARI
    for(n=1,1e3,if(!isprime(6*n-5)&&!isprime(6*n-1),print1(n", "))) \\ Charles R Greathouse IV, Nov 10 2011
Showing 1-7 of 7 results.

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