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.

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A259487 Least positive integer m with prime(m)+2 and prime(prime(m))+2 both prime such that prime(m*n)+2 and prime(prime(m*n))+2 are both prime.

Original entry on oeis.org

2, 1860, 408, 25011, 51312, 37977, 695, 4071, 10970, 3621, 17671, 12005, 1230, 19494, 542, 577, 408, 2476, 584, 542, 469, 34229, 343, 24078, 3011, 25749, 20706, 24198, 2478, 3926, 1030, 1030, 13857, 3621, 343, 13380, 2476, 4922, 2476, 296, 19176, 29175, 34737, 13, 625, 2956, 408, 572, 7, 469, 15604, 9699, 26515, 2167, 5302, 9773, 54254, 1410, 4524, 4351
Offset: 1

Views

Author

Zhi-Wei Sun, Jun 28 2015

Keywords

Comments

Conjecture: Any positive rational number r can be written as m/n with m and n terms of A259488.
This implies that there are infinitely many primes p with p+2 and prime(p)+2 both prime.
I have verified the conjecture for all those r = a/b with a,b = 1,...,400. - Zhi-Wei Sun, Jun 29 2015

Examples

			a(1) = 2 since prime(2)+2 = 3+2 = 5 and prime(prime(2))+2 = prime(3)+2 = 7 are both prime, but prime(1)+2 = 4 is composite.
a(49) = 7 since prime(7)+2 = 17+2 = 19, prime(prime(7))+2 = prime(17)+2 = 59+2 = 61, prime(49*7)+2 = 2309+2 = 2311 and prime(prime(49*7))+2 = prime(2309)+2 = 20441+2 = 20443 are all prime.

References

  • Zhi-Wei Sun, Problems on combinatorial properties of primes, in: M. Kaneko, S. Kanemitsu and J. Liu (eds.), Number Theory: Plowing and Starring through High Wave Forms, Proc. 7th China-Japan Seminar (Fukuoka, Oct. 28 - Nov. 1, 2013), Ser. Number Theory Appl., Vol. 11, World Sci., Singapore, 2015, pp. 169-187.

Links

Crossrefs

Cf. A000040, A001359, A006512, A236458, A258803, A258836, A259488, A259492.

Programs

  • Mathematica
    PQ[k_]:=PrimeQ[Prime[k]+2]&&PrimeQ[Prime[Prime[k]]+2]
Do[k=0;Label[bb];k=k+1;If[PQ[k]&&PQ[n*k], Goto[aa], Goto[bb]];Label[aa];Print[n," ", k];Continue,{n,1,60}]

A261533 Primes p such that p+2 is prime with prime(p+2)-prime(p)=6.

Original entry on oeis.org

3, 5, 59, 2789, 5231, 6947, 8087, 11717, 15269, 16229, 17207, 17909, 18059, 18131, 24917, 28751, 35279, 37307, 39227, 39239, 41201, 43787, 45821, 47741, 51869, 53087, 53609, 58439, 64577, 69857, 70919, 75707, 79631, 84869, 92381, 93479, 96179, 102197, 102929, 106187
Offset: 1

Views

Author

Zhi-Wei Sun, Aug 23 2015

Keywords

Comments

The conjecture in A261528 implies that the current sequence has infinitely many terms.
Note that for each n > 2 the difference prime(n+2)-prime(n) is at least 6.

Examples

			a(1) = 3 since 3 and 3+2 = 5 are twin prime, and prime(5)-prime(3) = 11-5 = 6.
a(2) = 5 since 5 and 5+2 = 7 are twin prime, and prime(7)-prime(5) = 17-11 = 6.

References

  • Zhi-Wei Sun, Problems on combinatorial properties of primes, in: M. Kaneko, S. Kanemitsu and J. Liu (eds.), Number Theory: Plowing and Starring through High Wave Forms, Proc. 7th China-Japan Seminar (Fukuoka, Oct. 28 - Nov. 1, 2013), Ser. Number Theory Appl., Vol. 11, World Sci., Singapore, 2015, pp. 169-187.

Links

Crossrefs

Cf. A000040, A001359, A006512, A023201, A046117, A236458, A259488, A259539, A261528.

Programs

  • Mathematica
    f[n_]:=Prime[n]
PQ[k_]:=PrimeQ[f[k]+2]&&f[f[k]+2]-f[f[k]]==6
n=0;Do[If[PQ[k],n=n+1;Print[n," ",f[k]]],{k,1,10119}]
Select[Partition[Prime[Range[11000]],2,1],#[[2]]-#[[1]]==2&&Prime[#[[1]]+ 2]- Prime[#[[1]]]==6&][[All,1]] (* Harvey P. Dale, Apr 26 2020 *)
  • PARI
    isok(i)=p=prime(i);isprime(p+2)&&prime(p+2)-prime(p)==6;
first(m)=my(v=vector(m));i=1;for(j=1,m,while(!isok(i),i++);v[j]=prime(i);i++);v; \\ Anders Hellström, Aug 23 2015
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