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A069457 Lowest primes in twin packs.

%I A069457 #25 Mar 02 2025 21:54:11
%S A069457 3,101,137,179,419,809,1019,1049,1481,1871,1931,2081,2111,2969,3251,
%T A069457 3359,3461,4217,4259,5009,5651,5867,6689,6761,6947,7331,7547,8219,
%U A069457 8969,9419,10007,11057,11159,11699,12239,13001,13709,13997
%N A069457 Lowest primes in twin packs.
%C A069457 As the example (below) explains, "A twin pack of primes contains 2 or more pairs of twin primes, between which pairs there are no other primes." The key phrase is "or more." The first twin pack is therefore ((3,5),(5,7),(11,13),(17,19)). Because all of the consecutive primes from 3 to 19 are included in this twin pack, the lowest primes in the two pairs of twin primes ((5,7),(11,13)) and ((11,13),(17,19)) are not included because they are already subsumed in the first twin pack. - _Harvey P. Dale_, Mar 02 2025
%H A069457 Robert Israel, <a href="/A069457/b069457.txt">Table of n, a(n) for n = 1..10000</a>
%e A069457 A twin pack of primes contains 2 or more pairs of twin primes, between which pairs there are no other primes. 137 is in the sequence because 137,139 are primes and the next primes are 149,151.
%p A069457 state:= 0: p:=13: Res:= 3: count:= 1;
%p A069457 while count < 100 do
%p A069457   q:= nextprime(p);
%p A069457   if state = 0 then
%p A069457      if q = p+2 then state:= 1; r:= p; p:= nextprime(q);
%p A069457      else p:= q
%p A069457      fi;
%p A069457    elif state = 1 then
%p A069457      if q = p+2 then
%p A069457        count:= count+1; Res:= Res, r; state:= 2; p:= nextprime(q);
%p A069457      else p:= q; state:= 0
%p A069457      fi
%p A069457    else
%p A069457      if q = p+2 then
%p A069457        p:= nextprime(q);
%p A069457      else p:= q; state:= 0
%p A069457      fi
%p A069457    fi
%p A069457 od:
%p A069457 Res; # _Robert Israel_, Jan 13 2020
%Y A069457 Cf. A001359, A048614, A053778.
%K A069457 nonn
%O A069457 1,1
%A A069457 _Neil Fernandez_, Mar 23 2002