cp's OEIS Frontend

A273756 Least p for which min { x >= 0 | p + (2n+1)*x + x^2 is composite } reaches the (local) maximum given in A273770.

%I A273756 #37 Feb 16 2025 08:33:35
%S A273756 41,43,47,53,61,71,83,97,113,131,151,173,197,223,251,281,313,347,383,
%T A273756 421,461,503,547,593,641,691,743,797,73303,73361,73421,73483,3443897,
%U A273756 3071069,3071137,15949847,76553693,365462323,365462399,2204597,9721,1842719,246407633,246407719,246407807,246407897,246407989
%N A273756 Least p for which min { x >= 0 | p + (2n+1)*x + x^2 is composite } reaches the (local) maximum given in A273770.
%C A273756 All terms are prime, since this is necessary and sufficient to get a prime for x = 0.
%C A273756 The values given in A273770 are the number of consecutive primes obtained for x = 0, 1, 2, ....
%C A273756 Sequence A273595 is the subsequence of terms for which 2n+1 is prime.
%C A273756 For even coefficients of the linear term, the answer would always be q=2, the only choice that yields a prime for x=0 and also for x=1 if (coefficient of the linear term)+3 is prime.
%C A273756 The initial term a(n=0) = 41 corresponds to Euler's famous prime-generating polynomial 41+x+x^2. Some subsequent terms are equal to the primes this polynomial takes for x=1,2,3,.... This stems from the fact that adding 2 to the coefficient of the linear term is equivalent to shifting the x-variable by 1. Since here we require x >= 0, we find a reduced subset of the previous sequence of primes, missing the first one, starting with q equal to the second one. (It is known that there is no better prime-generating polynomial of this form than Euler's, see the MathWorld page and A014556. "Better" means a larger p producing p-1 primes in a row. However, the prime k-tuple conjecture suggests that there should be arbitrarily long runs of primes of this form (for much larger p), i.e., longer than 41, but certainly much less than the respective p. Therefore we speak of local maxima.)
%H A273756 Don Reble, <a href="/A273756/b273756.txt">Table of n, a(n) for n = 0..100</a>
%H A273756 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/Prime-GeneratingPolynomial.html">Prime-Generating Polynomial</a>
%H A273756 Index to the OEIS, <a href="/index/Pri">Entries related to primes produced by polynomials</a>.
%o A273756 (PARI) A273756(n,p=2*n+1,L=10^(5+n\10),m=0,Q)={forprime(q=1,L, for(x=1,oo, ispseudoprime(q+p*x+x^2)&& next; x>m&& [Q=q,m=x]; break));Q}
%Y A273756 Cf. A273595, A273770.
%Y A273756 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.
%Y A273756 The first line of data coincides with that of A202018, A107448, A155884 (and also A140755, A142719, except for some initial terms), which are all related.
%K A273756 nonn
%O A273756 0,1
%A A273756 _M. F. Hasler_, May 26 2016
%E A273756 Edited, following a remark by _Don Reble_, by _M. F. Hasler_, Jan 23 2018
%E A273756 a(27) corrected and more terms from _Don Reble_, Feb 15 2018