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Let q_k= = p(*(p+2) be product of k-th pair of twin primes; sequence gives values of p+2 such that (q_k)^2 > q_{k-i}}*q_{k+i} for all 1 <= i <= k-1.

5, 13, 31, 61, 103, 139, 181, 193, 229, 421, 523, 571, 601, 811, 823, 1021, 1231, 1279, 1291, 1609, 1669, 1873, 2083, 2551, 2659, 2689, 2971, 3121, 3253, 3331, 3361, 3769, 3823, 3919, 4003, 5233, 5419, 5479, 6091, 6271, 6553, 6661, 6691, 8221, 8821, 8971

Even if there are infinitely many twin primes, it is not clear that this sequence is infinite. The Hardy-Littlewood conjecture implies that there are infinitely many twin primes where p+2 is not in the sequence. _. - _Robert Israel_, Apr 02 2014

E.g. ((11*13)^2 > (5*7)*(17*19): (11*13)^2 > (3*5)*(29*31)).

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Even if there are infinitely many twin primes, it is not clear that this sequence is infinite. The Hardy-Littlewood conjecture implies that there are infinitely many twin primes where p+2 is not in the sequence. Robert Israel, Apr 02 2014

N:= 20000:

Primes:= [seq(ithprime(i), i=1..N)]:

Twink:= select(t-> (Primes[t+1]=Primes[t]+2), [$1..N-1]):

Qk:= [seq(Primes[i]*Primes[i+1], i=Twink)]:

filter:= proc(k)

local T, i;

T:= Qk[k]^2;

for i from 1 to k-1 do

if Qk[k-i]*Qk[k+i]>=T then return false fi

od;

true

end;

R:= select(filter, [$1 .. floor(nops(Twink)/2)]):

A021007:= map(k -> Primes[Twink[k]+1], R); # Robert Israel, Apr 02 2014

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easy,nonn,changed

Charles R Greathouse IV, <a href="/A021007/b021007.txt">Table of n, a(n) for n = 1..10000</a>

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