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A140118
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Extrapolation for (n + 1)-st odd prime made by fitting least-degree polynomial to first n odd primes.
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2
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3, 7, 9, 19, 3, 49, -39, 151, -189, 381, -371, 219, 991, -4059, 11473, -26193, 53791, -100639, 175107, -281581, 410979, -506757, 391647, 401587, -2962157, 9621235, -24977199, 57408111, -120867183, 236098467, -428880285, 719991383, -1096219131, 1442605443, -1401210665, 99178397, 4340546667
(list; graph; refs; listen; history; internal format)
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OFFSET
| 1,1
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COMMENTS
| 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).
a(n) = sum_1_n p_i (-1)^(n - i) binomial(n, i - 1) where p_i are the primes.
Can anything be said about the pattern of positive and negative values?
How many of these terms are the correct (n + 1)th prime?
How many terms are prime?
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).
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LINKS
| Jonathan Wellons, Table of n, a(n) for n = 1..1500
Jonathan Wellons, Home Page.
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EXAMPLE
| 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.
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CROSSREFS
| Cf. A140119.
Sequence in context: A079464 A036976 A031273 * A191106 A110674 A003528
Adjacent sequences: A140115 A140116 A140117 * A140119 A140120 A140121
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KEYWORD
| sign
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AUTHOR
| Jonathan Wellons (wellons(AT)gmail.com), May 08 2008, May 19 2008
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