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.

A158866 Indices of twin primes if the sum of these twin primes+1 is an upper twin prime.

Original entry on oeis.org

2, 5, 30, 31, 66, 73, 88, 91, 141, 147, 217, 513, 607, 637, 743, 760, 784, 845, 856, 911, 920, 938, 949, 958, 994, 1015, 1031, 1092, 1150, 1246, 1373, 1470, 1553, 1586, 1768, 1814, 1871, 2017, 2029, 2129, 2261, 2271, 2331, 2370, 2458, 2488, 2510, 2545, 2579
Offset: 1

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Author

Cino Hilliard, Mar 28 2009

Keywords

Comments

If the sum is a member of a twin prime pair, it always is the upper twin prime member. [Proof: Twin primes are sequentially of the form 6n-1 and 6n+1. Then adding according to the condition, we get 6n-1+6n+1+1 = 12n+1. This implies 12n+1 is an upper member since if it were a lower member, 12n+1+2 would be the upper member but 12n+3 is not prime contradicting the definition of a twin prime. Therefore 12n+1 must be an upper twin prime member as stated.]

Examples

			The 30th lower twin prime is 659. 659+661+1 = 1321, prime and 1319 is too.
Then 1319 is the lower member of the twin prime pair (1319,1321). So 30 is in the sequence.
		

Crossrefs

Cf. A158870.

Programs

  • Maple
    count:= 0: Res:= NULL:
    k:= 1:
    for j from 1 while count < 100 do
      if isprime(6*j-1) and isprime(6*j+1) then
        k:= k+1;
        if isprime(12*j-1) and isprime(12*j+1) then
           count:= count+1;
           Res:= Res,k;
        fi
      fi
    od:
    Res; # Robert Israel, Mar 06 2018
  • Mathematica
    utpQ[{a_, b_}]:=And@@PrimeQ[a + b + {1, -1}]; Flatten[Position[Select[ Partition[Prime[Range[25000]],2,1],#[[2]]-#[[1]]==2&],?utpQ]] (* _Harvey P. Dale, Sep 16 2013 *)
  • PARI
    twinl(n) = { local(c,x); c=0; x=1; while(c
    				

Formula

{k: A054735(k)+1 = A006512(j), any j} - R. J. Mathar, Apr 06 2009

Extensions

Edited by R. J. Mathar, Apr 06 2009