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

Author

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

Keywords

• sign

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 prime(n + 1).

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).

Examples

			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.
		

Links

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

Crossrefs

Cf. A140119.

Programs

PARI

a(n) = sum(i=1, n, prime(i+1)*(-1)^(n-i)*binomial(n, i-1)); \\ Michel Marcus, Jul 07 2025

Formula

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