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

%I A158866 #19 Mar 07 2018 02:31:58
%S A158866 2,5,30,31,66,73,88,91,141,147,217,513,607,637,743,760,784,845,856,
%T A158866 911,920,938,949,958,994,1015,1031,1092,1150,1246,1373,1470,1553,1586,
%U A158866 1768,1814,1871,2017,2029,2129,2261,2271,2331,2370,2458,2488,2510,2545,2579
%N A158866 Indices of twin primes if the sum of these twin primes+1 is an upper twin prime.
%C A158866 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.]
%H A158866 Robert Israel, <a href="/A158866/b158866.txt">Table of n, a(n) for n = 1..10000</a>
%F A158866 {k: A054735(k)+1 = A006512(j), any j} - _R. J. Mathar_, Apr 06 2009
%e A158866 The 30th lower twin prime is 659. 659+661+1 = 1321, prime and 1319 is too.
%e A158866 Then 1319 is the lower member of the twin prime pair (1319,1321). So 30 is in the sequence.
%p A158866 count:= 0: Res:= NULL:
%p A158866 k:= 1:
%p A158866 for j from 1 while count < 100 do
%p A158866   if isprime(6*j-1) and isprime(6*j+1) then
%p A158866     k:= k+1;
%p A158866     if isprime(12*j-1) and isprime(12*j+1) then
%p A158866        count:= count+1;
%p A158866        Res:= Res,k;
%p A158866     fi
%p A158866   fi
%p A158866 od:
%p A158866 Res; # _Robert Israel_, Mar 06 2018
%t A158866 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 *)
%o A158866 (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
%o A158866 g(n)=for(x=1,n,y=2*twinl(x)+3;if(isprime(y)&&isprime(y-2), print1(x",")))
%Y A158866 Cf. A158870.
%K A158866 nonn
%O A158866 1,1
%A A158866 _Cino Hilliard_, Mar 28 2009
%E A158866 Edited by _R. J. Mathar_, Apr 06 2009