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A351921
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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.
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0
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2, 26, 21, 86, 33, 1238, 4401, 4586, 16161, 18561, 81, 37046, 85478, 180146, 339866
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OFFSET
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2,1
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COMMENTS
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LINKS
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MATHEMATICA
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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 *)
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PROG
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(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
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CROSSREFS
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KEYWORD
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nonn,more
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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