A140118 Extrapolation for (n + 1)-st odd prime made by fitting least-degree polynomial to first n odd primes.

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

Offset

1,1

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).

Links

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

Jonathan Wellons, Home Page.

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.

Crossrefs

Cf. A140119.

Sequence in context: A230116 A361005 A031273 * A191106 A324699 A110674

Adjacent sequences: A140115 A140116 A140117 * A140119 A140120 A140121

Keyword

sign

Author

Jonathan Wellons (wellons(AT)gmail.com), May 08 2008, May 19 2008

Status

approved