A309285
a(n) is the smallest odd composite k such that prime(n)^((k-1)/2) == 1 (mod k) and q^((k-1)/2) == -1 (mod k) for every prime q < prime(n).
Original entry on oeis.org
341, 29341, 48354810571, 493813961816587, 32398013051587
Offset: 1
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isok(n,k) = (k%2==1) && !isprime(k) && Mod(prime(n), k)^((k-1)/2) == Mod(1, k) && !forprime(q=2, prime(n)-1, if(Mod(q, k)^((k-1)/2) != Mod(-1, k), return(0)));
a(n) = for(k=9, oo, if(isok(n, k), return(k))); \\ Daniel Suteu, Jul 22 2019
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.
Original entry on oeis.org
2, 26, 21, 86, 33, 1238, 4401, 4586, 16161, 18561, 81, 37046, 85478, 180146, 339866
Offset: 2
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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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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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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])
Showing 1-2 of 2 results.
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