A273595 Least q > 0 such that min { x >= 0 | q + prime(n)*x + x^2 is composite } is a (local) maximum, cf. A273756 & A273770.

43, 47, 53, 71, 83, 113, 131, 173, 251, 281, 383, 461, 503, 593, 743, 73361, 73421, 3071069, 15949847, 76553693, 2204597, 1842719, 246407807, 986578883, 73975907, 4069235123, 1244414939, 25213427, 656856899, 30641069183, 8221946477, 41730358853, 10066886927, 285340609997, 6232338461

Offset: 2

Author

M. F. Hasler, May 26 2016

Keywords

• nonn

Comments

This is a subsequence of A273756 which considers all odd numbers (2n+1) instead of only prime(n) as coefficients of the linear term.

All terms are necessarily prime, since this is necessary and sufficient to get a prime for x = 0.

The respective minima (= number of consecutive primes for x = 0, 1, 2, ...) are given in A273597.

It has been pointed out by Don Reble that the prime k-tuple conjecture predicts infinitely long sequences of primes of the given form, therefore we consider the "local" maxima, for q below some appropriate (large) limit: see sequences A273756 & A273770 for further details. - M. F. Hasler, Feb 17 2020

Links

• Eric Weisstein's World of Mathematics, Prime-Generating Polynomial.
• Index to entries related to primes produced by polynomials.

Crossrefs

Cf. A273756, A273770.

Cf. also A002837 (n such that n^2-n+41 is prime), A007634 (n such that n^2+n+41 is composite), A005846 (primes of form n^2+n+41), A097823, A144051, A187057 .. A187060, A190800, A191456 ff.

Programs

PARI

A273595(n)=A273756(prime(n)\2) \\ changed Feb 17 2020

Formula

a(n) = A273756((prime(n) - 1)/2). - M. F. Hasler, Feb 17 2020

Extensions

Edited and extended using A273756(0..100) due to Don Reble, by M. F. Hasler, Feb 17 2020