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 1-8 of 8 results.

A125821 Numbers k for which 8*k+5 and 8*k+7 are twin primes.

Original entry on oeis.org

3, 12, 18, 24, 33, 57, 102, 132, 153, 159, 162, 234, 243, 249, 267, 279, 288, 297, 318, 348, 423, 432, 444, 447, 477, 489, 519, 528, 552, 564, 579, 627, 684, 687, 717, 774, 783, 837, 858, 918, 948, 969, 984, 993
Offset: 1

Views

Author

Artur Jasinski, Dec 10 2006

Keywords

Comments

From Zak Seidov, Apr 19 2008: (Start)
Proof that all numbers in this sequence are divisible by 3:
if n=(3k+1), then 8n+7=8(3k+1)+7=3(5+8 k) (composite)
if n=(3k+2), then 8n+5=8(3k+2)+5=3(7+8 k) (composite),
so if we require that both 8n+5 and 8n+7 are primes, then n=3k, hence all terms in this sequence are multiples of 3. QED. (End)

Links

Crossrefs

Cf. A001109.
For a(n)/3 see A139404.
Cf. A125822, A139402, A139404.

Programs

  • Mathematica
    Do[If[PrimeQ[8n + 5] && PrimeQ[8n + 7], Print[n]], {n, 1, 1000}]
Select[Range[3,6000,3],AllTrue[8#+{5,7},PrimeQ]&] (* The program uses the AllTrue function from Mathematica version 10 *) (* Harvey P. Dale, Sep 14 2018 *)

A139405 Numbers k such that 8*k+1 and 8*k+7 are primes.

Original entry on oeis.org

2, 5, 9, 12, 24, 29, 32, 44, 54, 57, 74, 75, 80, 107, 110, 122, 129, 137, 152, 162, 165, 170, 179, 185, 194, 200, 207, 219, 222, 234, 249, 260, 267, 285, 297, 299, 302, 305, 332, 339, 362, 414, 432, 452, 470, 500, 509, 519, 555, 557, 564, 570, 582, 584, 599
Offset: 1

Views

Author

Artur Jasinski, Apr 19 2008

Keywords

Links

Crossrefs

Cf. A017077, A004771.
Cf. A125821, A125822, A139402, A139404.

Programs

  • Mathematica
    a = {}; Do[If[PrimeQ[8 n + 1] && PrimeQ[8 n + 7], AppendTo[a, n]], {n, 1, 1000}]; a

A139406 Numbers k such that 8*k+1 and 8*k+5 are primes.

Original entry on oeis.org

12, 24, 39, 57, 84, 96, 117, 126, 162, 186, 201, 234, 249, 267, 297, 309, 327, 336, 354, 357, 369, 402, 432, 441, 459, 462, 474, 516, 519, 564, 591, 621, 654, 696, 711, 717, 732, 777, 822, 942, 969, 984, 1011, 1029, 1086, 1092, 1116, 1167, 1179, 1272, 1341
Offset: 1

Views

Author

Artur Jasinski, Apr 19 2008

Keywords

Links

Crossrefs

Cf. A125821, A125822, A139402, A139404, A139405, A139407.

Programs

  • Mathematica
    a = {}; Do[If[PrimeQ[8 n + 1] && PrimeQ[8 n + 5], AppendTo[a, n]], {n, 1, 1000}]; a
Select[Range[1500],And@@PrimeQ[8 #+{1,5}]&] (* Harvey P. Dale, Aug 14 2011 *)

A139407 Numbers k such that 8*k+3 and 8*k+7 are primes.

Original entry on oeis.org

2, 5, 8, 20, 38, 47, 62, 80, 92, 107, 110, 113, 185, 197, 233, 260, 275, 293, 317, 332, 335, 338, 377, 395, 398, 488, 500, 653, 668, 722, 740, 755, 818, 863, 905, 950, 962, 965, 1052, 1055, 1067, 1097, 1100, 1193, 1202, 1217, 1223, 1235, 1262, 1280, 1283
Offset: 1

Views

Author

Artur Jasinski, Apr 19 2008

Keywords

Links

Crossrefs

Cf. A125821, A125822, A139402, A139404, A139405, A139406.

Programs

  • Mathematica
    a = {}; Do[If[PrimeQ[8 n + 3] && PrimeQ[8 n + 7], AppendTo[a, n]], {n, 1, 1000}]; a
Select[Range[1500],And@@PrimeQ[8 # +{3,7}]&] (* Harvey P. Dale, Mar 24 2011 *)

A139533 Numbers k such that numbers 24*k + 11 and 24*k + 13 are twin primes.

Original entry on oeis.org

0, 2, 4, 7, 9, 14, 17, 27, 34, 42, 45, 59, 60, 67, 69, 74, 80, 84, 94, 97, 135, 137, 140, 144, 147, 160, 167, 170, 177, 189, 199, 205, 212, 235, 244, 255, 262, 277, 282, 284, 289, 300, 304, 305, 314, 342, 345, 349, 359, 367, 375, 392, 419, 420, 422, 430, 434, 437
Offset: 1

Views

Author

Artur Jasinski, Apr 25 2008

Keywords

Links

Crossrefs

Numbers k such that:
24*k - 1 and 24*k + 1 are twin primes, see A137920.
24*k + 5 and 24*k + 7 are twin primes, see A139404.
24*k + 17 and 24*k + 19 are twin primes, see A139534.

Programs

  • Magma
    [k:k in [0..450]|IsPrime(24*k + 11) and IsPrime(24*k + 13)]; // Marius A. Burtea, Dec 31 2019
  • Mathematica
    a = {}; Do[If[PrimeQ[24 n + 11] && PrimeQ[24 n + 13], AppendTo[a, n]], {n, 0, 2000}]; a

A139534 Numbers k such that numbers 24*k + 17 and 24*k + 19 are twin primes.

Original entry on oeis.org

0, 1, 5, 11, 21, 23, 25, 26, 33, 35, 36, 43, 53, 61, 70, 71, 86, 88, 110, 113, 116, 123, 135, 138, 163, 166, 168, 175, 176, 180, 186, 193, 196, 208, 225, 226, 235, 243, 253, 268, 273, 278, 281, 310, 333, 355, 373, 376, 386, 410, 413, 435, 438, 453, 455, 460, 478
Offset: 1

Views

Author

Artur Jasinski, Apr 25 2008

Keywords

Links

Crossrefs

Cf. A137920, A139404, A139533.

Programs

  • Magma
    [k:k in [0..500]|IsPrime(24*k + 17) and IsPrime(24*k + 19)]; // Marius A. Burtea, Dec 31 2019
  • Mathematica
    a = {}; Do[If[PrimeQ[24 n + 17] && PrimeQ[24 n + 19], AppendTo[a, n]], {n, 0, 2000}]; a

A139403 Prime numbers k such that 8*k+3 and 8*k+5 are also primes.

Original entry on oeis.org

7, 13, 43, 103, 127, 181, 223, 241, 283, 421, 433, 733, 787, 853, 1291, 1303, 1531, 1567, 1741, 2017, 2161, 2281, 2593, 2857, 2953, 3163, 3361, 3571, 3673, 4003, 4051, 4441, 4513, 4597, 4663, 4831, 4903, 5503, 5647, 5923, 6067, 6091, 6217, 6361, 6427
Offset: 1

Views

Author

Artur Jasinski, Apr 19 2008

Keywords

Links

Crossrefs

Cf. A125822, A139402, A139404.
Subsequence of prime terms of A124192.

Programs

  • Mathematica
    a = {}; Do[If[PrimeQ[8 n + 5] && PrimeQ[8 n + 3] && PrimeQ[n],AppendTo[a, n]], {n, 1, 10000}]; a
Select[Prime[Range[1000]],AllTrue[8#+{3,5},PrimeQ]&] (* Requires Mathematica version 10 or later *) (* Harvey P. Dale, Apr 30 2020 *)

A303550 Numbers k such that abs(60*k^2 - 1710*k + 12150) +- 1 are twin primes.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 16, 17, 18, 19, 20, 27, 33, 34, 35, 36, 38, 41, 50, 56, 57, 64, 66, 69, 75, 81, 85, 86, 90, 93, 98, 103, 106, 119, 121, 133, 136, 141, 143, 146, 150, 181, 182, 189, 195, 202, 207, 208, 212, 215, 218, 219, 225
Offset: 1

Views

Author

Amiram Eldar, Apr 26 2018

Keywords

Comments

The formula was discovered by Andrew T. Gazsi in 1961.
The polynomial can also be given as 30*(2*k - 27)*(k - 15). Its value is negative (-30) at k = 14 and 0 and k = 15.
Beiler erroneously claimed that the polynomial generates twin primes for k = 1 to 20.

Examples

			1 is in the sequence since 60*1^2 - 1710*1 + 12150 = 10500 and (10499, 10501) are twin primes.

References

  • Albert H. Beiler, Recreations in the Theory of Numbers: The Queen of Mathematics Entertains, 2nd ed., Dover Publications, Inc., New York, 1966, p. 225.
  • Joseph B. Dence and Thomas P. Dence, Elements of the Theory of Numbers, Academic Press, 1999, problem 1.94, p.35.
  • Andrew T. Gazsi, A Formula to Generate Prime Pairs, Recreational Mathematics Magazine, edited by Joseph S. Madachy, Issue 6, December 1961, p. 44.

Links

Crossrefs

Cf. A001097, A001359, A006512, A088485, A108897, A124518, A124519, A139404, A185660.

Programs

  • Maple
    filter:= proc(n) local k;
  k:= abs(60*n^2-1710*n+12150);
  isprime(k+1) and isprime(k-1)
end proc:
select(filter, [$1..300]); # Robert Israel, Jun 19 2018
  • Mathematica
    f[n_] := 60n^2 - 1710n + 12150; aQ[n_]:=PrimeQ[f[n]-1] && PrimeQ[f[n]+1]; Select[Range[225], aQ]
Select[Range[250],AllTrue[Abs[60#^2-1710#+12150+{1,-1}],PrimeQ]&] (* Harvey P. Dale, May 17 2025 *)
  • PARI
    f(n) = abs(60*n^2 - 1710*n + 12150);
isok(n) = my(fn=f(n)); isprime(fn-1) && isprime(fn+1); \\ Michel Marcus, Apr 27 2018
Showing 1-8 of 8 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

Content is available under The OEIS End-User License Agreement.