A126193 - OEIS

%I #10 Sep 16 2024 06:35:52

%S 5,17,29,41,59,71,107,137,179,191,197,227,239,269,281,311,347,419,431,

%T 461,569,599,617,641,659,809,821,827,857,881,1019,1049,1061,1091,1151,

%U 1229,1277,1289,1301,1319,1427,1451,1481,1487,1607,1619,1667,1697,1721

%N 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 p = q-k^4+k^2 where p and q are lesser of twin primes and p < q.

%C May be connected with the twin prime conjecture (see link).

%H Robert Israel, <a href="/A126193/b126193.txt">Table of n, a(n) for n = 1..10000</a>

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/TwinPrimeConjecture.html">Twin Prime Conjecture</a>

%e 5 = 2^2+1 and 17 = 2^4+1; 5 and 17 are lesser of twin primes;

%e 41 = 4^2+25 and 281 = 4^4+25; 41 and 281 are lesser of twin primes.

%p filter:= proc(p) local s,k;

%p   if not(isprime(p) and isprime(p+2)) then return false fi;

%p   for k from 2 do

%p     s:= p - k^2;

%p     if s < 0 then return false fi;

%p     if isprime(s+k^4) and isprime(s+k^4+2) then return true fi;

%p   od

%p end proc:

%p select(filter, [seq(i,i=5..2000, 6)]); # _Robert Israel_, Sep 15 2024

%o (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 Cf. A001359, A126769, A126194.

%K easy,nonn

%O 1,1

%A _Tomas Xordan_, Mar 07 2007

%E Edited and checked by _Klaus Brockhaus_, Mar 09 2007

%E Definition corrected by _Robert Israel_, Sep 15 2024