A140119 Extrapolation for (n + 1)-st prime made by fitting least-degree polynomial to first n primes.

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

Offset: 1

Author

Jonathan Wellons (wellons(AT)gmail.com), May 08 2008

Keywords

• sign

Comments

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?

Row sums of triangle A095195. - Reinhard Zumkeller, Oct 10 2013

Examples

			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.
		

Links

• Jonathan Wellons, Table of n, a(n) for n = 1..1500

Crossrefs

Cf. A082594, A140118.

Programs

Haskell

a140119 = sum . a095195_row  -- Reinhard Zumkeller, Oct 10 2013

PARI

a(n) = sum(i=1, n, prime(i)*(-1)^(n-i)*binomial(n, i-1)); \\ Michel Marcus, Jun 28 2020

Formula

a(n) = Sum_{i=1..n} prime(i) * (-1)^(n-i) * C(n,i-1).