A349644 Array read by antidiagonals, n >= 2, m >= 0: T(n,m) is the smallest prime p = prime(k) such that all n-th differences of (prime(k), ..., prime(k+n+m)) are zero.

3, 251, 17, 9843019, 347, 347, 121174811, 2903, 2903, 41

Offset: 2

Author

Pontus von Brömssen, Nov 23 2021

Keywords

• nonn
• tabl
• hard
• more

Comments

T(n,m) = prime(k), where k is the smallest positive integer such that A095195(j,n) = 0 for k+n <= j <= k+n+m.

Equivalently, T(n,m) 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+m.

Examples

			Array begins:
  n\m|   0       1           2           3           4
  ---+------------------------------------------------
  2  |   3     251     9843019   121174811           ?
  3  |  17     347        2903       15373      128981
  4  | 347    2903       15373      128981    19641263
  5  |  41    8081      128981    19641263   245333213
  6  | 211  128981    19641263   245333213   245333213
  7  | 271  386471    81028373   245333213 27797667517
  8  |  23 2022971   245333213 27797667517           ?
  9  | 191 7564091 10246420463           ?           ?
		

Crossrefs

Cf. A006560 (row n=2), A349642 (row n=3), A349643 (column m=0).

Cf. A095195.

Programs

Python

from sympy import nextprime
def A349644(n,m):
    d = [float('inf')]*(n-1)
    p = [0]*(n+m)+[2]
    c = 0
    while 1:
        del p[0]
        p.append(nextprime(p[-1]))
        d.insert(0,p[-1]-p[-2])
        for i in range(1,n):
            d[i] = d[i-1]-d[i]
        if d.pop() == 0:
            if c == m: return p[0]
            c += 1
        else:
            c = 0

Formula

T(n,m) <= T(n-1,m+1).

T(n,m) <= T(n, m+1).

Sum_{j=0..n} (-1)^j*binomial(n,j)*prime(k+i+j) = 0 for 0 <= i <= m, where prime(k) = T(n,m).