A352597 a(n) is the smallest k > 1 such that k^n + 1 has all prime divisors p == 1 (mod n).

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.

Links

Table of n, a(n) for n=1..57.

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

Adjacent sequences: A352594 A352595 A352596 * A352598 A352599 A352600

Keyword

nonn,hard

Author

Kevin P. Thompson, Mar 21 2022

Status

approved