A021007 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

Offset: 1

Author

Naohiro Nomoto

Keywords

• nonn

Comments

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

Examples

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

Links

• Charles R Greathouse IV, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A028388, A021005.

Programs

Maple

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

PARI

twins=List(); p=3;forprime(q=5,1e5,if(q-p==2,listput(twins,q)); p=q); for(k=1,(#twins+1)\2, for(i=1,k-1,if(twins[k]^2 < twins[k-i]*twins[k+i],next(2))); print1(twins[k]", ")) \\ Charles R Greathouse IV, Apr 02 2014

Extensions

a(1) inserted by Robert Israel, Apr 02 2014