%I #6 Dec 20 2021 18:40:42

%S 23,2,25771,74159,245333129,245333113

%N a(n) is the smallest prime p such that the Hankel matrix of the 2*n-1 consecutive primes starting at p is singular; a(n) = 0 if no such p exists.

%C a(n) is the k-th prime, where k is the smallest positive integer such that A350200(n,k) = 0.

%C For a(n) = prime(k), a nontrivial linear relation c_1*prime(j) + ... + c_n*prime(j+n-1) = 0 holds for k <= j <= k+n-1. The vector (c_1, ..., c_n) is in the kernel of the Hankel matrix of (prime(k), ..., prime(k+2*n-2)). (Such a relation always holds for k <= j <= k+n-2, starting with an arbitrary sequence in place of the primes.)

%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Hankel_matrix">Hankel matrix</a>

%e Example

%e | | | vector in the kernel

%e n | a(n) | primepi(a(n)) | of the Hankel matrix

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

%e 3 | 23 | 9 | (7, 3, -8)

%e 4 | 2 | 1 | (6, -3, -2, 1)

%e 5 | 25771 | 2838 | (1, -2, 2, -2, 1)

%e 6 | 74159 | 7315 | (1, -2, 1, 1, -2, 1)

%e 7 | 245333129 | 13437898 | (0, 0, 0, 1, -3, 3, -1)

%e 8 | 245333113 | 13437897 | (0, 0, 0, 0, 1, -3, 3, -1)

%e For n = 3, the relation 7*prime(j) + 3*prime(j+1) - 8*prime(j+2) = 0 holds for 9 <= j <= 11, i.e.,

%e 7*23 + 3*29 - 8*31 = 0,

%e 7*29 + 3*31 - 8*37 = 0,

%e 7*31 + 3*37 - 8*41 = 0.

%e The ten prime gaps following prime(13437901) = 245333213 are 20, 18, 16, 14, 12, 10, 8, 6, 4, 2 (see A349642). This gives both a(7) = prime(13437898) and a(8) = prime(13437897).

%o (Python)

%o from sympy import prime,nextprime,Matrix

%o def A350201(n):

%o p = [prime(j) for j in range(1,2*n)]

%o while Matrix(n,n,lambda i,j:p[i+j]).det():

%o del p[0]

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

%o return p[0]

%Y Cf. A349642, A349643, A350200.

%K nonn,more

%O 3,1

%A _Pontus von Brömssen_, Dec 19 2021