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
Programs
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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 *) -
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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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])
Extensions
a(15)-a(16) from Jon E. Schoenfield, Mar 01 2022
Comments