A273770 Max { min { x >= 0 | p + (2*n+1)*x + x^2 is composite }, p < 10^(5+n/10) }.

40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 24, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 13, 12, 11, 10, 11, 12, 11, 12, 12, 13, 12, 12, 13, 16, 17, 16, 15, 14, 13, 13, 12, 11, 12, 13, 13, 14, 13, 13, 13, 12, 13, 14, 13, 14, 15, 14, 14, 13, 14, 14, 13

Offset: 0

Author

M. F. Hasler, May 26 2016

Keywords

• nonn

Comments

The values for p are given in A273756 which is the main entry, see there for further information and (cross)references.

From the initial values, the sequence seems strictly decreasing, with a(n) = 40-n, however, this property does not persist beyond a(27) = 13.

The upper limit on p ensures that we have a well-defined sequence: The prime k-tuple conjecture predicts existence of arbitrarily long sequences of primes of the given form, and thus unbounded minimal value of x. However, the corresponding prime tuples are expected to appear for much larger values of p. The given limit should be understood as "below the first/next such prime tuple", and in general the values a(n) should not change if that limit would be increased by some orders of magnitude. There might be counterexamples, which would be interesting. The given limit was chosen for lack of a more natural expression, and is relatively small. It could be replaced by a more appropriate function of n if a proposal is available, which should not affect the values given so far. - M. F. Hasler, Jan 22 2018, edited Feb 17 2020

Links

• M. F. Hasler, Table of n, a(n) for n = 0..100 (using A273756(0..100) from Don Reble), Feb 17 2020
• Index to sequences related to primes produced by polynomials.

Crossrefs

Cf. A273595, A273756.

Cf. also A002837, A007634, A005846, A097823, A144051, A187057 ... A187060, A190800, A191456 ff.

Programs

PARI

{A273770(n, p=2*n+1, L=10^(5+n/10), m)=forprime(q=1,L, for(x=1,oo, ispseudoprime(q+p*x+x^2) || (x>m && !m=x) || break));m}

Formula

a(n) = 40 - n for 0 <= n <= 27.

Extensions

Corrected and extended by Don Reble, Feb 15 2018