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A140119
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Extrapolation for (n + 1)-st prime made by fitting least-degree polynomial to first n primes.
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5
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2, 4, 8, 8, 22, -6, 72, -92, 266, -426, 838, -1172, 1432, -398, -3614, 15140, -41274, 95126, -195698, 370876, -652384, 1063442, -1570116, 1961852, -1560168, -1401888, 11023226, -36000318, 93408538, -214275608, 450374202, -879254356, 1599245876, -2695464868, 4138070460, -5539280974
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OFFSET
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1,1
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COMMENTS
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Construct the least-degree polynomial p(x) which fits the first n primes (p has degree n-1 or less). Then predict the next prime by evaluating p(n+1).
Can anything be said about the pattern of positive and negative values?
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LINKS
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FORMULA
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a(n) = Sum_{i=1..n} prime(i) * (-1)^(n-i) * C(n,i-1).
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EXAMPLE
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The lowest-order polynomial having points (1,2), (2,3), (3,5) and (4,7) is f(x) = 1/6 (-x^3 +9x^2 -14x +18). When evaluated at x = 5, f(5) = 8.
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PROG
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(Haskell)
(PARI) a(n) = sum(i=1, n, prime(i)*(-1)^(n-i)*binomial(n, i-1)); \\ Michel Marcus, Jun 28 2020
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CROSSREFS
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KEYWORD
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sign
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AUTHOR
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Jonathan Wellons (wellons(AT)gmail.com), May 08, 2008
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STATUS
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approved
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