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 1-4 of 4 results.

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

A178504 Numbers n such that n^2 + 13 is an emirp.

Original entry on oeis.org

2, 10, 12, 18, 44, 60, 88, 108, 110, 114, 116, 122, 192, 198, 282, 380, 446, 574, 588, 604, 612, 618, 838, 840, 864, 970, 1032, 1068, 1104, 1148, 1186, 1228, 1258, 1314, 1368, 1384, 1390, 1412, 1754, 1888, 1894, 1930, 2658, 2660, 2728, 2784, 2804
Offset: 1

Views

Author

Eva-Maria Zschorn (e-m.zschorn(AT)zaschendorf.km3.de), May 29 2010

Keywords

Comments

A decimal emirp/mirp ("prime" / (German) "prim", spelled backwards) is defined as a prime number p whose reversal R(p) is also prime, but which is not a palindromic prime.

Examples

			2^2 + 13 = 17 = prime(7), 71 = prime(20), so 2 is in the sequence.
10^2 + 13 = 113 = prime(30), 311 = prime(64), so 10 is in the sequence.
28^2 + 13 = 797, which is a palindromic prime, so 28 is not in the sequence.

References

  • W. W. R. Ball, H. S. M. Coxeter: Mathematical Recreations and Essays, 13th edition, Dover Publications, 2010
  • H. Steinhaus: Kaleidoskop der Mathematik, VEB Dt. Verl. d. Wissenschaften, Berlin, 1959

Links

Crossrefs

Subsequence of A113536.
Cf. A138375, A176978, A178044.

Programs

  • Mathematica
    fQ[n_] := If[ PrimeQ[n^2 + 13], Block[{id = IntegerDigits[n^2 + 13]}, rid = Reverse@ id; PrimeQ@ FromDigits@ rid && rid != id]]; Select[ Range@ 3000, fQ] (* Robert G. Wilson v, Jul 26 2010 *)

Extensions

More terms from Robert G. Wilson v, Jul 26 2010

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

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 1-4 of 4 results.

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