A273595 - OEIS

%I #38 Feb 16 2025 08:33:35

%S 43,47,53,71,83,113,131,173,251,281,383,461,503,593,743,73361,73421,

%T 3071069,15949847,76553693,2204597,1842719,246407807,986578883,

%U 73975907,4069235123,1244414939,25213427,656856899,30641069183,8221946477,41730358853,10066886927,285340609997,6232338461

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

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

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

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

%C 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

%H Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/Prime-GeneratingPolynomial.html">Prime-Generating Polynomial</a>.

%H <a href="/index/Pri">Index to entries related to primes produced by polynomials</a>.

%F a(n) = A273756((prime(n) - 1)/2). - _M. F. Hasler_, Feb 17 2020

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

%Y Cf. A273756, A273770.

%Y 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.

%K nonn,changed

%O 2,1

%A _M. F. Hasler_, May 26 2016

%E Edited and extended using A273756(0..100) due to _Don Reble_, by _M. F. Hasler_, Feb 17 2020