OFFSET
2,1
COMMENTS
Equivalently, a(n) is the smallest prime p = prime(k) such that there is a polynomial f of degree at most n-1 such that f(j) = prime(j) for k <= j <= k+n.
a(n) = prime(k), where k is the smallest positive integer such that A095195(k+n,n) = 0.
FORMULA
Sum_{j=0..n} (-1)^j*binomial(n,j)*prime(k+j) = 0, where prime(k) = a(n).
EXAMPLE
The first six consecutive primes for which the fifth difference is 0 are (41, 43, 47, 53, 59, 61), so a(5) = 41. The successive differences are (2, 4, 6, 6, 2), (2, 2, 0, -4), (0, -2, -4), (-2, -2), and (0).
MATHEMATICA
With[{prs=Prime[Range[10^6]]}, Table[SelectFirst[Partition[prs, n+1, 1], Differences[#, n]=={0}&][[1]], {n, 2, 21}]] (* The program generates the first 20 terms of the sequence. *) (* Harvey P. Dale, Aug 10 2024 *)
PROG
(Python)
from math import comb
from sympy import nextprime
def A349643(n):
plist, clist = [2], [1]
for i in range(1, n+1):
plist.append(nextprime(plist[-1]))
clist.append((-1)**i*comb(n, i))
while True:
if sum(clist[i]*plist[i] for i in range(n+1)) == 0: return plist[0]
plist = plist[1:]+[nextprime(plist[-1])] # Chai Wah Wu, Nov 25 2021
CROSSREFS
KEYWORD
nonn
AUTHOR
Pontus von Brömssen, Nov 23 2021
STATUS
approved