A386207 Numbers m > 1 such that there exists k such that k | m, k^k = k mod m and 1 < k < m.

6, 10, 12, 14, 15, 18, 20, 21, 22, 24, 26, 28, 30, 33, 34, 36, 38, 39, 40, 42, 44, 45, 46, 48, 50, 51, 52, 54, 55, 56, 57, 58, 60, 62, 63, 65, 66, 68, 69, 70, 72, 74, 75, 76, 78, 80, 82, 84, 85, 86, 87, 88, 90, 91, 92, 93, 94, 95, 98, 100, 102, 104, 105, 106, 108, 110, 111, 112, 114, 116

Offset: 1

Author

Arjun Maneesh Agarwal, Jul 15 2025

Keywords

• nonn

Comments

The sequence seems to be very similar to A324455 but there are terms present here, which are absent there and vice versa. Also, the k satisfying the property for a given n, even when it exists, is different for both.

The sequence is infinite as A016825 (4m+2, m > 0) is a subsequence with the corresponding k = 2m + 1.

Another subsequence is A002997 (Carmichael numbers).

Examples

			6 is a term as 3^3 = 27 = 3 mod 6.
10 is a term as 5^5 = 3125 = 5 mod 10.
		

Links

• Arjun Maneesh Agarwal, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A324455, A016825, A002997.

Programs

Haskell

divisors n = [x | x <- [2..n], n `mod` x == 0]
property n = any (\x -> x^x `mod` n == x) $ divisors n
inRange t = [x | x <- [2..t], property x]

PARI

isok(m) = fordiv(m, k, if((k>1) && (kMichel Marcus, Jul 16 2025