%I A100873
%S A100873 645,1105,2701,2821,4681,6601,10261,12801,14491,16705,18721,19951,25761,
%T A100873 29341,30121,31609,33153,39865,41041,42799,49141,52633,55245,62745,
%U A100873 68101,72885,83665,85489,90751,104653,107185,129889,129921
%N A100873 Pseudotwinprimes: primes p such that p+2 divides p^(p+2)+2 and p+2 is 
               composite.
%C A100873 Conjecture 1: If p and p+2 are prime (twin primes), then p+2 divides 
               p^(p+2)+2. Compared to the 1517 twin primes less than 130000, there 
               were 33 pseudoprime occurrences. Conjecture 2: If for a randomly 
               chosen prime p, p+2 divides p^(p+2)+2, then there is a greater than 
               98% chance that p and p+2 are twin primes. The sequence also contains 
               several Carmichael numbers. In addition, If we relax the condition 
               that p is prime or just odd, we get A001567 341,561,645,1105,1387,
               1729,1905,2047.. Sarrus numbers.
%F A100873 For primes p if p+2 divides p^(p+2) + 2 then p+2 is likley to be prime. 
               If p+2 is composite, then p+2 is a pseudotwinprime.
%e A100873 For prime p = 643, 645 divides 643^(645)+ 2 and 645 is composite.
%o A100873 (PARI) twtotwp2(n,n2,k) = { local(x,y,x2,c); c=0; forprime(x=n,n2, x2=x+2; 
               y=x^x2+k; if(y%x2==0&!isprime(x2),c++; print1(x+2","); ); ); print(); 
               print(c","pitwin(n2)) } pitwins(n) = \The number of twin prime pairs 
               <= n. { local(c,x); c=0; forprime(x=3,n, if(isprime(x+2),c++) ); 
               return(c) }
%Y A100873 Sequence in context: A135384 A061324 A089295 this_sequence A063844 A067845 
               A057942
%Y A100873 Adjacent sequences: A100870 A100871 A100872 this_sequence A100874 A100875 
               A100876
%K A100873 hard,nonn
%O A100873 3,1
%A A100873 Cino Hilliard (hillcino368(AT)gmail.com), Jan 09 2005