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%I #3 Nov 11 2010 07:34:06
%S 3,7,9,19,3,49,-39,151,-189,381,-371,219,991,-4059,11473,-26193,53791,
%T -100639,175107,-281581,410979,-506757,391647,401587,-2962157,9621235,
%U -24977199,57408111,-120867183,236098467,-428880285,719991383,-1096219131,1442605443,-1401210665,99178397,4340546667
%N Extrapolation for (n + 1)-st odd prime made by fitting least-degree polynomial to first n odd primes.
%C Construct the least-degree polynomial p(x) which fits the first n odd primes (p has degree n - 1 or less). Then predict the next prime by evaluating p(n + 1).
%C a(n) = sum_1_n p_i (-1)^(n - i) binomial(n, i - 1) where p_i are the primes.
%C Can anything be said about the pattern of positive and negative values?
%C How many of these terms are the correct (n + 1)th prime?
%C How many terms are prime?
%C The terms at indices 1, 2, 4, 5, 8, 13, 17, 20, 24, 32, 54, 75, 105, 283, 676, 769, 1205 and 1300 actually are prime (ignoring negative signs).
%H Jonathan Wellons, <a href="/A140118/b140118.txt">Table of n, a(n) for n = 1..1500</a>
%H Jonathan Wellons, <a href="http://jonathanwellons.com/">Home Page</a>.
%e The lowest-order polynomial having points (1,3), (2,5), (3,7) and (4,11) is f(x) = 1/3 (x^3 - 6x^2 + 17x - 3). When evaluated at x = 5, f(5) = 19.
%Y Cf. A140119.
%K sign
%O 1,1
%A Jonathan Wellons (wellons(AT)gmail.com), May 08 2008, May 19 2008
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