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URL: https://oeis.org/A131626

⇱ A131626 - OEIS

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A131626
Prime subsequences in the evaluation of the first derivatives of quadratic equations with positive successive prime coefficients.
0
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.
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
KEYWORD
nonn
AUTHOR
Cino Hilliard, Sep 03 2007
STATUS
approved