A351921 a(n) is the smallest nonzero number k such that gcd(prime(1)^k + 1, prime(2)^k + 1, ..., prime(n)^k + 1) > 1 and gcd(prime(1)^k + 1, prime(2)^k + 1, ..., prime(n+1)^k + 1) = 1.

2, 26, 21, 86, 33, 1238, 4401, 4586, 16161, 18561, 81, 37046, 85478, 180146, 339866

Offset

2,1

Comments

Apparently, a(n) = (A307965(n+1) + 1)/2 - 1 for n>=3. - Hugo Pfoertner, Mar 02 2022

Links

Table of n, a(n) for n=2..16.

Mathematica

a[n_] := Module[{k = 1, p = Prime[Range[n + 1]]}, While[GCD @@ (Most[p]^k + 1) == 1 || GCD @@ (p^k + 1) > 1, k++]; k]; Array[a, 10, 2] (* Amiram Eldar, Feb 26 2022 *)

Prog

(Python)
from sympy import sieve
from math import gcd
from functools import reduce
sieve.extend_to_no(50)
pr = list(sieve._list)
terms = [0]*100
for i in range(2, 85478+1):
    k, g, len_f = 1, 2, 0
    while g != 1:
        k += 1
        len_f += 1
        g = reduce(gcd, [t**i + 1 for t in pr[:k]])
    if len_f > 1 and terms[len_f] == 0:
        terms[len_f] = i
print(terms[2:15])
(PARI) isok(k, n) = my(v = vector(n+1, i, prime(i)^k+1)); (gcd(v) == 1) && (gcd(Vec(v, n)) != 1);
a(n) = my(k=1); while (!isok(k, n), k++); k; \\ Michel Marcus, Mar 18 2022

Crossrefs

Cf. A213052, A241043, A307965.

Sequence in context: A035420 A241202 A278479 * A374183 A022375 A072663

Adjacent sequences: A351918 A351919 A351920 * A351922 A351923 A351924

Keyword

nonn,more

Author

Gleb Ivanov, Feb 25 2022

Extensions

a(15)-a(16) from Jon E. Schoenfield, Mar 01 2022

Status

approved