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

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

Sequence in context: A364442 A334456 A293343 * A086762 A296305 A349997

Adjacent sequences: A131623 A131624 A131625 * A131627 A131628 A131629

Keyword

nonn

Author

Cino Hilliard, Sep 03 2007

Status

approved