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A352597
a(n) is the smallest k > 1 such that k^n + 1 has all prime divisors p == 1 (mod n).
0
2, 2, 6, 2, 10, 6, 28, 2, 18, 10, 22, 6, 52, 14, 60, 2, 102, 36, 190, 20, 756, 66, 46, 18, 2550, 26, 2970, 28, 58, 120, 310, 2, 330, 170, 11550, 6, 148, 570, 156, 140, 82, 2184, 172, 88, 3040020, 184, 282, 42, 7252, 110, 7548, 312, 106, 1440, 41800, 42, 11172
OFFSET
1,1
COMMENTS
Equivalently, a(n) is the smallest k > 1 such that for all divisors d of k^n + 1, d == 1 (mod n).
A298299 is a subsequence.
All terms in this sequence are even since for odd k the expression k^n + 1 is divisible by 2 which is not congruent to 1 (mod n) for any n > 1.
If n is odd, a(n)^n + 1 is divisible by a(n) + 1. Therefore, a(n) + 1 == 1 (mod n) and so n | a(n) for odd n.
Theorem: a(n) = 2 if and only if n is a power of 2.
FORMULA
a(2n) = A298299(n).
EXAMPLE
a(3) = 6 since 6^3 + 1 = 217 = 7 * 31 and both factors are congruent to 1 (mod 3).
PROG
(PARI) isok(k, n) = my(f=factor(k^n+1)); for (i=1, #f~, if (Mod(f[i, 1], n) != 1, return(0))); return(1);
a(n) = my(k=2); while (!isok(k, n), k+=2); k; \\ Michel Marcus, Mar 22 2022
CROSSREFS
Cf. A298076, A298299 (bisection), A298310, A298398.
Sequence in context: A232625 A099985 A298299 * A110765 A176991 A264666
KEYWORD
nonn,hard
AUTHOR
Kevin P. Thompson, Mar 21 2022
STATUS
approved