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.

A051779 Primes of form pq + 2 where p and q are twin primes.

Original entry on oeis.org

17, 37, 22501, 32401, 57601, 72901, 176401, 324901, 1664101, 1742401, 5336101, 6502501, 7452901, 11289601, 11492101, 18147601, 21622501, 34222501, 34574401, 40449601, 45968401, 81000001, 85377601, 92736901, 110880901, 118592101
Offset: 1

Views

Author

Joe DeMaio (jdemaio(AT)kennesaw.edu), Dec 09 1999

Keywords

Comments

Starting with 3rd term, 22501, all terms are of the form 900n^2+1 with n=5, 6, 8, 9, 14, 19, 43, 44, 77, 85 (A125251). - Zak Seidov, Dec 07 2008
Primes of the form (p^2 + q^2)/2, where p and q are twin primes. - Thomas Ordowski and Altug Alkan, Mar 19 2017

Examples

			The third term 22501 is a member of the sequence because 22501=149*151+2, 22501 is prime and {149,151} is a twin prime pair.

Links

Crossrefs

Cf. A048880, A051779, A000040, A001359, A005384, A006512, A037074, A054735.
Cf. A125251. - Zak Seidov, Dec 07 2008

Programs

  • Maple
    with (numtheory): for n from 1 to 2000 do if (ithprime(n+1)-ithprime(n)=2) then if (tau(ithprime(n)*ithprime(n+1)+2)=2) then print(ithprime(n),ithprime(n+1), ithprime(n)*ithprime(n+1)+2); fi; fi; od;
  • Mathematica
    lst={};Do[p=Prime[n];If[Length[Divisors[p-2]]==4&&(Divisors[p-2][[3]]-Divisors[p-2][[2]])==2, AppendTo[lst, p]], {n, 6*10^5}];lst (* Vladimir Joseph Stephan Orlovsky, Aug 08 2008 *)
Select[(First[#]Last[#]+2)&/@Select[Partition[Prime[Range[2700]], 2,1], Last[#]-First[#]==2&],PrimeQ]  (* Harvey P. Dale, Mar 11 2011 *)
Select[2 + Times @@@ Select[ Partition[ Prime@ Range@ 1350, 2, 1], First[#] + 2 == Last[#] &], PrimeQ] (* Robert G. Wilson v, Mar 12 2001 *)

Formula

{A037074(k) + 2} INTERSECT {A000040}. {A001359(k) * A006512(k) + 2} INTERSECT {A000040}. {A054735(k)^2 + 2*A054735(k) + 2} INTERSECT {A000040}. - Jonathan Vos Post, May 11 2006

Extensions

Edited by R. J. Mathar, Aug 08 2008

A138220 Numbers k such that 900*k^2 + 1 is prime.

Original entry on oeis.org

3, 4, 5, 6, 7, 8, 9, 10, 14, 19, 23, 25, 26, 31, 32, 36, 38, 43, 44, 45, 47, 48, 62, 64, 66, 77, 81, 82, 85, 90, 91, 92, 95, 108, 112, 113, 116, 122, 129, 130, 136, 138, 139, 142, 147, 151, 153, 155, 164, 178, 179, 181, 183, 185, 190, 192, 195, 196, 199, 201, 202, 204
Offset: 1

Views

Author

Zak Seidov, May 05 2008

Keywords

Comments

Includes all terms from A125251.
Conjecture: (30*k)^(2^m)+1 is prime for some numbers k at all values of m >= 0. - Richard R. Forberg, Feb 06 2021

Crossrefs

Cf. A125251, A111175, A341183.

Programs

A284045 Numbers k such that p = 6k-1 and q = 6k+1 are twin primes and (p^2 + q^2)/2 is prime.

Original entry on oeis.org

1, 25, 30, 40, 45, 70, 95, 215, 220, 385, 425, 455, 560, 565, 710, 775, 975, 980, 1060, 1130, 1500, 1540, 1605, 1755, 1815, 1995, 2280, 2305, 2335, 2425, 2705, 2775, 3010, 3020, 3090, 3190, 3230, 3600, 3640, 3895, 3945, 4455, 4480, 4615, 4900, 5045, 5495, 5525, 5750, 5880
Offset: 1

Views

Author

Thomas Ordowski and Altug Alkan, Mar 19 2017

Keywords

Comments

Numbers k such that (6k)^2 + 1 is prime and p = 6k-1 and q = 6k+1 are twin primes.
Numbers k such that p*q + 2 is prime, where p = 6k-1 and q = 6k+1 are twin primes.

Links

Crossrefs

Subsequence of A002822.
Cf. A051779.

Programs

  • Mathematica
    fQ[n_] := AllTrue[{30n -1, 30n +1, 900n^2 +1}, PrimeQ]; Join[{1},5*Select[ Range@1200, fQ]] (* Robert G. Wilson v, Mar 19 2017 *)
  • PARI
    print1(1, ", "); for(n=1, 15000, if(isprime(30*n-1)&& isprime(30*n+1)&& isprime(900*n^2+1), print1(5*n, ", ")));

Formula

a(n) == 0 (mod 5) for n > 1.
a(n+1) = 5 * A125251(n).
Showing 1-3 of 3 results.

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