A131626 - OEIS

A131626

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

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OFFSET

1,1

COMMENTS

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

LINKS

Table of n, a(n) for n=1..57.

FORMULA

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,..

EXAMPLE

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 6th position.

PROG

(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) ) ) }

CROSSREFS