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COMMENTS
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a(n) is the k-th prime, where k is the smallest positive integer such that A350200(n,k) = 0.
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.)
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EXAMPLE
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Example
| | | vector in the kernel
n | a(n) | primepi(a(n)) | of the Hankel matrix
--+-----------+---------------+------------------------------
3 | 23 | 9 | (7, 3, -8)
4 | 2 | 1 | (6, -3, -2, 1)
5 | 25771 | 2838 | (1, -2, 2, -2, 1)
6 | 74159 | 7315 | (1, -2, 1, 1, -2, 1)
7 | 245333129 | 13437898 | (0, 0, 0, 1, -3, 3, -1)
8 | 245333113 | 13437897 | (0, 0, 0, 0, 1, -3, 3, -1)
For n = 3, the relation 7*prime(j) + 3*prime(j+1) - 8*prime(j+2) = 0 holds for 9 <= j <= 11, i.e.,
7*23 + 3*29 - 8*31 = 0,
7*29 + 3*31 - 8*37 = 0,
7*31 + 3*37 - 8*41 = 0.
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).
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PROG
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(Python)
from sympy import prime, nextprime, Matrix
p = [prime(j) for j in range(1, 2*n)]
while Matrix(n, n, lambda i, j:p[i+j]).det():
del p[0]
p.append(nextprime(p[-1]))
return p[0]
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