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 1-5 of 5 results.

A176978 Numbers n such that n^2 + 13 and n^2 + 13^2 are primes.

Original entry on oeis.org

2, 10, 12, 28, 42, 60, 68, 108, 110, 122, 142, 180, 198, 200, 268, 282, 340, 378, 380, 418, 430, 488, 500, 502, 530, 612, 742, 788, 802, 838, 840, 912, 942, 948, 952, 1010, 1080, 1102, 1148, 1232, 1270, 1428, 1452, 1472, 1502, 1522, 1538, 1720, 1778, 1830
Offset: 1

Views

Author

Eva-Maria Zschorn (e-m.zschorn(AT)zaschendorf.km3.de), Apr 30 2010

Keywords

Comments

Numbers are subsequence of A113536 and A176969
See comments and references of A176969

Examples

			2^2 + 13 = 17 = prime(7), 2^2 + 13^2 = 173 = prime(40), 2 is first term.

Links

Crossrefs

Cf. A000040, A000290, A056899, A113536, A176371, A176969.

Programs

  • Mathematica
    Select[Range[2000],AllTrue[#^2+{13,13^2},PrimeQ]&] (* The program uses the AllTrue function from Mathematica version 10 *) (* Harvey P. Dale, Sep 28 2015 *)
  • PARI
    isok(n) = isprime(n^2 + 13) && isprime(n^2 + 13^2); \\ Michel Marcus, Aug 27 2013

A177833 Numbers k such that k^2 - 13 and k^2 + 13 are primes.

Original entry on oeis.org

4, 12, 18, 72, 84, 114, 198, 354, 378, 588, 612, 618, 864, 912, 948, 1032, 1068, 1134, 1320, 1410, 1428, 1452, 1500, 1830, 1956, 2046, 2058, 2172, 2298, 2448, 2634, 2748, 2844, 2856, 3192, 3246, 3390, 3474, 3846, 3906, 4092, 4182, 4506, 4842, 4884, 4890
Offset: 1

Views

Author

Eva-Maria Zschorn (e-m.zschorn(AT)zaschendorf.km3.de), May 14 2010

Keywords

Examples

			4^2 - 13 = 3 = prime(2), 4^2 + 13 = 29 = prime(10).
12^2 - 13 = 131 = prime(32), 12^2 + 13 = 157 = prime(37).
948^2 - 13 = 898691 = prime(71194), 948^2 + 13 = prime(71195), first case that they are consecutive primes.

References

  • J. Matousek and J. Nesetril, Diskrete Mathematik: eine Entdeckungsreise, Springer-Lehrbuch, 2. Aufl., Berlin, 2007

Links

Crossrefs

Cf. A138375, A154648, A176969, A176978, A248785.

Programs

  • Magma
    [n: n in [4..1000]| IsPrime(n^2-13) and IsPrime(n^2+13)]; // Vincenzo Librandi, Nov 30 2010
  • Maple
    with(numtheory): A248785:=n->`if`(isprime(n^2-13) and isprime(n^2+13), n, NULL): seq(A248785(n), n=1..10^4); # Wesley Ivan Hurt, Oct 13 2014
  • Mathematica
    Select[Range[2,5000,2],AllTrue[#^2+{13,-13},PrimeQ]&] (* Harvey P. Dale, May 28 2024 *)

Extensions

More terms from Vincenzo Librandi, May 16 2010
Name edited by Michel Marcus, Nov 25 2024

A178652 Primes of the form 4^k + 13^2.

Original entry on oeis.org

173, 233, 1193, 16553, 262313, 67109033, 1073741993, 4611686018427388073, 73786976294838206633, 19807040628566084398385987753, 1361129467683753853853498429727072845993
Offset: 1

Views

Author

Eva-Maria Zschorn (e-m.zschorn(AT)zaschendorf.km3.de), Jun 01 2010

Keywords

Comments

Necessarily k is odd, 4^2 + 13^2 a multiple of 5.

Examples

			a(1) = 4^1 + 13^2 = 173.
a(2) = 4^3 + 13^2 = 233.
a(11) = 4^65 + 13^2 = 1361129467683753853853498429727072845993.
a(12) = 4^99 + 13^2 = 401734511064747568885490523085290650630550748445698208825513.

Crossrefs

Cf. A176969, A176978, A177833.

Programs

  • Mathematica
    Select[4^Range[0,70]+13^2,PrimeQ] (* Harvey P. Dale, Mar 05 2015 *)
  • PARI
    forstep(n=1,999,2,if(ispseudoprime(t=4^n+169),print1(t", "))) \\ Charles R Greathouse IV, Aug 27 2013

Extensions

New name from Charles R Greathouse IV, Aug 27 2013

A178639 Numbers m such that all three values m^2 + 13^k, k = 1, 2, 3, are prime.

Original entry on oeis.org

10, 12, 200, 268, 340, 418, 488, 530, 838, 840, 1102, 1720, 1830, 2240, 2410, 2768, 3148, 3202, 3318, 3322, 3958, 4162, 4610, 5080, 5672, 5700, 5722, 5870, 6178, 6302, 6480, 7490, 8130, 8262, 8888, 9132, 9602, 9618, 10638
Offset: 1

Views

Author

Ulrich Krug (leuchtfeuer37(AT)gmx.de), May 31 2010

Keywords

Comments

Subsequence of A176969.
The least-significant digit of all terms is one of 0, 2 or 8, because for odd digits m^2 + 13^k would be even (not prime), and for digits 4 and 6 the number m^2 + 13^2 is a multiple of 5.

Examples

			m=10 is in the sequence because 10^2 + 13 = 113 = prime(30), 10^2 + 13^2 = 269 = prime(57), 10^2 + 13^3 = 2297 = prime(342).
m=8888 is in the sequence because 8888^2 + 13 = 78996557 = prime(4614261), 8888^2 + 13^2 = 78996713 = prime(4614269), 8888^2 + 13^3 = 78998741 = prime(4614379).
m=6480 yields a prime 6480^2 + 13^k even for k=0.
m=7490 yields a prime 7490^2 + 13^k even for k=0 and k=4.

References

  • B. Bunch: The Kingdom of Infinite Number: A Field Guide, W. H. Freeman, 2001.
  • R. Courant, H. Robbins: What Is Mathematics? An Elementary Approach to Ideas and Methods, Oxford University Press, 1996.
  • G. H. Hardy, E. M. Wright, E. M., An Introduction to the Theory of Numbers (5th edition), Oxford University Press, 1980.

Crossrefs

Cf. A000040, A000290, A055394, A056899, A086380, A113536, A176371, A176969.

Extensions

keyword:base removed by R. J. Mathar, Jul 13 2010

A178653 Numbers k that 4^k + 13^2 is prime.

Original entry on oeis.org

1, 3, 5, 7, 9, 13, 15, 31, 33, 47, 65, 99, 103, 147, 197, 203, 257, 399, 411, 471, 497, 979, 1189, 2851, 3221, 4689, 5027, 7131, 7545, 9049, 9849
Offset: 1

Views

Author

Eva-Maria Zschorn (e-m.zschorn(AT)zaschendorf.km3.de), Jun 01 2010

Keywords

Comments

See A178652.

Examples

			4^1 + 13^2 = 173 = prime(40), 1 is first term.
4^3 + 13^2 = 233 = prime(51), 3 is 2nd term.
4^5 + 13^2 = 1193 = prime(196), 5 is 3rd term.
4^147 + 13^2 = 318286...15753 (89 digits), 147 is 14th term.
4^197 + 13^2 = 403...9753 (119 digits), 197 is 15th term.

Crossrefs

Cf. A176969, A176978, A177833, A178652.

Programs

  • Mathematica
    fQ[n_] := PrimeQ[(2^k)^2 + 169]; k = 1; lst = {}; While[k < 10^4, If[ fQ@k, AppendTo[lst, k]; Print@k]; k += 2]; lst (* Robert G. Wilson v, Jul 31 2010 *)
  • PARI
    forstep(k=1,999,2,if(ispseudoprime(4^n+169),print1(n", "))) \\ Charles R Greathouse IV, Aug 27 2013

Extensions

a(16)-a(31) from Robert G. Wilson v, Jul 31 2010
New name from Charles R Greathouse IV, Aug 27 2013
Showing 1-5 of 5 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.