editing

approved

So the subsequence 19,29,47,67,101 is in the sequence beginning in the 6-th 6th position.

approved

editing

_Cino Hilliard (hillcino368(AT)hotmail.com), _, Sep 03 2007

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 for 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,..

nonn,new

nonn

Prime subsequences in the evaluation of the first derivatives of quadratic equations with positive successive prime coefficients.

7, 11, 17, 11, 17, 19, 29, 47, 67, 101, 23, 31, 47, 43, 47, 71, 59, 89, 67, 101, 167, 71, 107, 79, 83, 103, 107, 127, 191, 317, 131, 197, 139, 151, 227, 163, 167, 251, 179, 269, 191, 199, 211, 317, 223, 227, 239, 359, 251, 263, 271, 283, 307, 461, 311, 467, 331

1,1

Conjecture: The number of primes in a row for f'(x,p) = 2*prime(p)*x + prime(p+1) is finite.

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 for 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,..

For x = 4 we have 2*prime(p)*x + prime(p+1) =

2*2*4+3 = 19 prime,

2*3*4+5 = 29 prime,

2*5*4+7 = 47 prime,

2*7*4+11 = 67 prime,

2*11*4+13 = 101 prime,

2*13*4+17 = 121 not prime.

So the subsequence 19,29,47,67,101 is in the sequence beginning in the 6-th position.

(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) ) ) }

nonn

Cino Hilliard (hillcino368(AT)hotmail.com), Sep 03 2007

approved