

A158866


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


3



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
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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 6n1 and 6n+1. Then adding according to the condition, we get 6n1+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*j1) and isprime(6*j+1) then
k:= k+1;
if isprime(12*j1) 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(x1)) } \\ The nth lower twin prime
g(n)=for(x=1, n, y=2*twinl(x)+3; if(isprime(y)&&isprime(y2), 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



