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%I #5 Mar 13 2015 00:47:04
%S 7,11,17,11,17,19,29,47,67,101,23,31,47,43,47,71,59,89,67,101,167,71,
%T 107,79,83,103,107,127,191,317,131,197,139,151,227,163,167,251,179,
%U 269,191,199,211,317,223,227,239,359,251,263,271,283,307,461,311,467,331
%N Prime subsequences in the evaluation of the first derivatives of quadratic equations with positive successive prime coefficients.
%C Conjecture: The number of primes in a row for f'(x,p) = 2*prime(p)*x + prime(p+1) is finite.
%F Define prime(p) = the p-th prime number. The equation f(x,p) = prime(p)x^2 + prime(p+1)x + prime(p+2) is differentiated to get f'(x,p) = 2prime(p)x + prime(p+1). Then f'(x,p) is evaluated at p=1,2,.. for each x =1,2,.. until f'(x,p) is not prime at which point x is incremented and p=1,2,..
%e For x = 4 we have 2*prime(p)*x + prime(p+1) =
%e 2*2*4+3 = 19 prime,
%e 2*3*4+5 = 29 prime,
%e 2*5*4+7 = 47 prime,
%e 2*7*4+11 = 67 prime,
%e 2*11*4+13 = 101 prime,
%e 2*13*4+17 = 121 not prime.
%e So the subsequence 19,29,47,67,101 is in the sequence beginning in the 6th position.
%o (PARI) g(n) = { local(x,y,p); for(x=0,n, for(p=1,n, y=2*prime(p)*x+prime(p+1); if(isprime(y),print1(y","),break) ) ) }
%K nonn
%O 1,1
%A _Cino Hilliard_, Sep 03 2007
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