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 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).
LINKS
Jonathan Wellons, Table of n, a(n) for n = 1..1500
FORMULA
a(n) = Sum_{i=1..n} prime(i+1)*(-1)^(n-i)*binomial(n, i-1).
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.
PROG
(PARI) a(n) = sum(i=1, n, prime(i+1)*(-1)^(n-i)*binomial(n, i-1)); \\ Michel Marcus, Jul 07 2025
CROSSREFS
KEYWORD
sign
AUTHOR
Jonathan Wellons (wellons(AT)gmail.com), May 08 2008, May 19 2008
STATUS
approved