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A341217 a(k) is the lesser of the first pair of twin primes that starts a chain of k pairs of twin primes (p(1),p(1)+2), ..., (p(k),p(k)+2) where p(j+1) = 5*p(j)+4.

%I A341217 #17 Jul 30 2022 23:58:23
%S A341217 3,11,5,1720949,22362444257,57703877539769
%N A341217 a(k) is the lesser of the first pair of twin primes that starts a chain of k pairs of twin primes (p(1),p(1)+2), ..., (p(k),p(k)+2) where p(j+1) = 5*p(j)+4.
%C A341217 a(5) > 10^8 if it exists.
%e A341217 3 and 5 are twin primes, but 5*3+4 = 19 and 21 are not twin primes.
%e A341217 11 and 13 are twin primes, 5*11+4 = 59 and 61 are twin primes, but 5*59+4 = 299 and 301 are not twin primes.
%e A341217 5 and 7 are twin primes, 5*5+4 = 29 and 31 are twin primes, 5*29+4 = 149 and 151 are twin primes, but 5*149+4 = 749 and 751 are not twin primes.
%e A341217 1720949 and 1720951 are twin primes, 5*1720949+4 = 8604749 and 8604751 are twin primes, 5*8604749+4 = 43023749 and 43023751 are twin primes,
%e A341217 5*8604749+4 = 215118749 and 215118751 are twin primes, 5*1075593749+4 = 1075593749 and 1075593751 are not twin primes.
%p A341217 V:= [3,0,0,0]:
%p A341217 count:= 1:
%p A341217 for p from 5 by 6 while count < 4 do
%p A341217   if isprime(p) and isprime(p+2) then
%p A341217     ct:= 1:
%p A341217     q:= p;
%p A341217     do
%p A341217       q:= 5*q+4;
%p A341217       if not (isprime(q) and isprime(q+2)) then break fi;
%p A341217       ct:= ct+1;
%p A341217     od;
%p A341217     if V[ct] = 0 then V[ct]:= p; count:= count+1; fi;
%p A341217   fi
%p A341217 od:
%p A341217 V;
%Y A341217 Cf. A001359, A174922.
%K A341217 nonn,more
%O A341217 1,1
%A A341217 _J. M. Bergot_ and _Robert Israel_, Feb 06 2021
%E A341217 a(5) from _Martin Ehrenstein_, Feb 07 2021
%E A341217 a(6) from _Martin Ehrenstein_, Feb 10 2021