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A126193 Lesser of twin primes (A001359) of the form p = k^2+s such that q = k^4+s is also a lesser of twin primes, q > p and s >= 0.

%I A126193 #11 Feb 16 2025 08:33:04
%S A126193 5,17,29,41,59,71,107,137,179,191,197,227,239,269,281,311,347,419,431,
%T A126193 461,569,599,617,641,659,809,821,827,857,881,1019,1049,1061,1091,1151,
%U A126193 1229,1277,1289,1301,1319,1427,1451,1481,1487,1607,1619,1667,1697,1721
%N A126193 Lesser of twin primes (A001359) of the form p = k^2+s such that q = k^4+s is also a lesser of twin primes, q > p and s >= 0.
%C A126193 p = q-k^4+k^2 where p and q are lesser of twin primes and p < q.
%C A126193 May be connected with the twin prime conjecture (see link).
%H A126193 Robert Israel, <a href="/A126193/b126193.txt">Table of n, a(n) for n = 1..10000</a>
%H A126193 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/TwinPrimeConjecture.html">Twin Prime Conjecture</a>
%e A126193 5 = 2^2+1 and 17 = 2^4+1; 5 and 17 are lesser of twin primes;
%e A126193 41 = 4^2+25 and 281 = 4^4+25; 41 and 281 are lesser of twin primes.
%p A126193 filter:= proc(p) local s,k;
%p A126193   if not(isprime(p) and isprime(p+2)) then return false fi;
%p A126193   for k from 2 do
%p A126193     s:= p - k^2;
%p A126193     if s < 0 then return false fi;
%p A126193     if isprime(s+k^4) and isprime(s+k^4+2) then return true fi;
%p A126193   od
%p A126193 end proc:
%p A126193 select(filter, [seq(i,i=5..2000, 6)]); # _Robert Israel_, Sep 15 2024
%o A126193 (PARI) {m=42; v=[]; for(k=2, m, for(s=1, (m+1)^2-1, if((p=k^2+s)<m^2&&isprime(p)&&isprime(p+2)&&(q=k^4+s)>p&&isprime(q)&&isprime(q+2), v=concat(v,p)))); v=listsort(List(v), 1); for(j=1, #v, print1(v[j], ","))} /* Klaus Brockhaus, Mar 09 2007 */
%Y A126193 Cf. A001359, A126769, A126194.
%K A126193 easy,nonn
%O A126193 1,1
%A A126193 _Tomas Xordan_, Mar 07 2007
%E A126193 Edited and checked by _Klaus Brockhaus_, Mar 09 2007
%E A126193 Definition corrected by _Robert Israel_, Sep 15 2024