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.

A116887 Primes p that remain prime through at least 3 iterations of function f(p)=p^2+4.

Original entry on oeis.org

5087, 11173, 16657, 47017, 54503, 185243, 207643, 300413, 306167, 341813, 391387, 849047, 1348577, 1438223, 1502923, 1857407, 1909267, 2121737, 2161163, 2288773, 2610133, 2725157, 2744723, 2779097, 2874463, 2881327, 3079927, 3149077, 3154483, 3173683, 3194483
Offset: 1

Views

Author

Giovanni Resta, Feb 27 2006

Keywords

Examples

			p=5087, f(p)= 25877573, f(f(p))= 669648784370333 and f(f(f(p)))= 448429494408664742387290530893 are all primes.

Links

Crossrefs

Cf. A062324, A116886, A116888, A116889.

Programs

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

A165218 Primes q of the form q=p^2+4 (p=prime) such that r=q^2+4 is also prime.

Original entry on oeis.org

13, 293, 10613, 18773, 76733, 97973, 458333, 552053, 1247693, 2647133, 4012013, 4592453, 11607653, 13520333, 20097293, 25877573, 34845413, 51509333, 53772893, 65399573, 65496653, 66373613, 72880373, 73496333, 86359853, 89737733
Offset: 1

Views

Author

Zak Seidov, Sep 08 2009

Keywords

Comments

Intersection of A062324 and A045637. Except of the first term, 13, all terms == 5 (mod 6) == 5 (mod 12) == 5 (mod 24) == 23 (mod 30)== 53 (mod 120). Values of primes p in A116886.

Examples

			Prime q=13=p^2+4 (p=3) and r=q^2+4=13^2+4=173 (prime).
Prime q=293=p^2+4 (p=17) and r=q^2+4=293^2+4=85853 (prime).

Links

Crossrefs

Cf. A045637, A062324, A116886.

Programs

  • Mathematica
    Reap[For[p = 2, p < 10^4, p = NextPrime[p], If[PrimeQ[q = p^2+4] && PrimeQ[q^2+4], Print[q]; Sow[q]]]][[2, 1]] (* Jean-François Alcover, Nov 07 2013 *)
Select[Prime[Range[2000]]^2+4,AllTrue[{#,#^2+4},PrimeQ]&] (* The program uses the AllTrue function from Mathematica version 10 *) (* Harvey P. Dale, Mar 13 2018 *)

Formula

a(n) = (A116886(n))^2 + 4.

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 1-4 of 4 results.

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