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A158866 Indices of twin primes if the sum of these twin primes+1 is an upper twin prime. 2
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,1

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.]

LINKS

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

FORMULA

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

EXAMPLE

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.

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 *)

PROG

(PARI) twinl(n) = { local(c, x); c=0; x=1; while(c<n, if(isprime(prime(x)+2), c++); x++; ); return(prime(x-1)) } \\ The n-th lower twin prime

g(n)=for(x=1, n, y=2*twinl(x)+3; if(isprime(y)&&isprime(y-2), print1(x", ")))

CROSSREFS

Cf. A158870.

Sequence in context: A086383 A118612 A187628 * A101078 A109739 A261414

Adjacent sequences:  A158863 A158864 A158865 * A158867 A158868 A158869

KEYWORD

nonn

AUTHOR

Cino Hilliard, Mar 28 2009

EXTENSIONS

Edited by R. J. Mathar, Apr 06 2009

STATUS

approved

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Last modified January 17 18:02 EST 2020. Contains 330987 sequences. (Running on oeis4.)