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 1-3 of 3 results.

A116886 Primes p that remain prime through at least 2 iterations of the function f(p) = p^2 + 4.

Original entry on oeis.org

3, 17, 103, 137, 277, 313, 677, 743, 1117, 1627, 2003, 2143, 3407, 3677, 4483, 5087, 5903, 7177, 7333, 8087, 8093, 8147, 8537, 8573, 9293, 9473, 10177, 10477, 11173, 13807, 14897, 15107, 16657, 19753, 21563, 22307, 24113, 26113, 26417, 26633
Offset: 1

Views

Author

Giovanni Resta, Feb 27 2006

Keywords

Comments

Numbers p with the property that p, q = p^2 + 4, and r = q^2 + 4 are all prime. - Zak Seidov, Sep 08 2009
a(n) = sqrt(A165218(n) - 4). - Zak Seidov, Sep 08 2009

Examples

			17 is prime, 17^2 + 4 = 293 is prime and 293^2 + 4 = 85853 is prime.

Links

Crossrefs

Cf. A062324, A116887, A116888, A116889, A045637, A062324, A165218.

Programs

  • Mathematica
    Select[Prime[Range[2*7! ]],PrimeQ[ #^2+4]&&PrimeQ[(#^2+4)^2+4]&] (* Vladimir Joseph Stephan Orlovsky, Feb 26 2010 *)
fQ[n_]:=AllTrue[Rest[NestList[#^2+4&,n,2]],PrimeQ]; Select[Prime[ Range[ 3000]],fQ] (* The program uses the AllTrue function from Mathematica version 10 *) (* Harvey P. Dale, Sep 21 2014 *)
  • PARI
    is(n)=my(q);isprime(p) && isprime(q=p^2+4) && isprime(q^2+4) \\ Charles R Greathouse IV, Nov 06 2013

Extensions

Edited by N. J. A. Sloane, Sep 18 2009 at the suggestion of R. J. Mathar

A116889 a(n) is the least prime p that remains prime through n iterations of function f(p)=p^2+4.

Original entry on oeis.org

2, 3, 3, 5087, 306167
Offset: 0

Views

Author

Giovanni Resta, Feb 27 2006

Keywords

Comments

The sequence is finite, since it can be proved that if p, f(p), f(f(p)), f(f(f(p))) and f(f(f(f(p)))) are all primes, then the next iteration gives a multiple of 13, greater than 13, thus a(k) for k>=5 does not exist.

Examples

			a(0)=2 since f(2)=8 is not prime. a(1)=a(2)=3 since both f(3)=13 and f(f(3))=173 are primes.

Crossrefs

Cf. A062324, A116886, A116887, A116888.

Extensions

Typo in Example fixed by Zak Seidov, Nov 07 2013

A116888 Primes p that remain prime through 4 iterations of the function f(p) = p^2 + 4.

Original entry on oeis.org

306167, 48639197, 64695713, 68252687, 87788237, 87813293, 160486967, 255974437, 283032247, 324609913, 361705873, 417684523, 449364197, 451995587, 454052213, 466037563, 536504713, 574746467, 596095613
Offset: 1

Views

Author

Giovanni Resta, Feb 27 2006

Keywords

Examples

			p = 306167, f(p) = 93738231893, f(f(p)) = 8786856118425842363453, f(f(f(p))) = 77208840445917661077402487029419236950083213 and the 88-digit number f(f(f(f(p)))) are all prime numbers.

Crossrefs

Cf. A062324, A116886, A116887, A116889.

Programs

  • Mathematica
    Select[Prime[Range[9! ]],PrimeQ[ #^2+4]&&PrimeQ[(#^2+4)^2+4]&&PrimeQ[((#^2+4)^2+4)^2+4]&&PrimeQ[(((#^2+4)^2+4)^2+4)^2+4]&] (* Vladimir Joseph Stephan Orlovsky, Feb 26 2010 *)
p4Q[p_]:=AllTrue[NestList[#^2+4&,p,4],PrimeQ]; Select[Prime[Range[312*10^5]],p4Q] (* Harvey P. Dale, Nov 20 2023 *)
Showing 1-3 of 3 results.

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

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