A349643 - OEIS

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A349643

Smallest prime p = prime(k) such that the n-th difference of (prime(k), ..., prime(k+n)) is zero.

3, 17, 347, 41, 211, 271, 23, 191, 33151, 541, 70891, 152681, 856637, 158047, 2010581, 24239, 7069423, 15149419, 9472693, 347957, 479691493, 994339579, 132480637, 4462552643, 1342424483, 4757283367, 20674291411, 21170786093, 9941224877, 68864319317, 8660066477

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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.

LINKS

Table of n, a(n) for n=2..32.

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