A349644 - OEIS

%I #12 Dec 05 2021 06:05:25

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

%N 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.

%C 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.

%C 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.

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

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

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

%e Array begins:

%e   n\m|   0       1           2           3           4

%e   ---+------------------------------------------------

%e   2  |   3     251     9843019   121174811           ?

%e   3  |  17     347        2903       15373      128981

%e   4  | 347    2903       15373      128981    19641263

%e   5  |  41    8081      128981    19641263   245333213

%e   6  | 211  128981    19641263   245333213   245333213

%e   7  | 271  386471    81028373   245333213 27797667517

%e   8  |  23 2022971   245333213 27797667517           ?

%e   9  | 191 7564091 10246420463           ?           ?

%o (Python)

%o from sympy import nextprime

%o def A349644(n,m):

%o     d = [float('inf')]*(n-1)

%o     p = [0]*(n+m)+[2]

%o     c = 0

%o     while 1:

%o         del p[0]

%o         p.append(nextprime(p[-1]))

%o         d.insert(0,p[-1]-p[-2])

%o         for i in range(1,n):

%o             d[i] = d[i-1]-d[i]

%o         if d.pop() == 0:

%o             if c == m: return p[0]

%o             c += 1

%o         else:

%o             c = 0

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

%Y Cf. A095195.

%K nonn,tabl,hard,more

%O 2,1

%A _Pontus von Brömssen_, Nov 23 2021