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

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

Author

Cino Hilliard, Mar 28 2009

Keywords

• nonn

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.
		

Links

• Robert Israel, Table of n, a(n) for n = 1..10000

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