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

%I A386207 #24 Jul 22 2025 22:14:35
%S A386207 6,10,12,14,15,18,20,21,22,24,26,28,30,33,34,36,38,39,40,42,44,45,46,
%T A386207 48,50,51,52,54,55,56,57,58,60,62,63,65,66,68,69,70,72,74,75,76,78,80,
%U A386207 82,84,85,86,87,88,90,91,92,93,94,95,98,100,102,104,105,106,108,110,111,112,114,116
%N A386207 Numbers m > 1 such that there exists k such that k | m, k^k = k mod m and 1 < k < m.
%C A386207 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.
%C A386207 The sequence is infinite as A016825 (4m+2, m > 0) is a subsequence with the corresponding k = 2m + 1.
%C A386207 Another subsequence is A002997 (Carmichael numbers).
%H A386207 Arjun Maneesh Agarwal, <a href="/A386207/b386207.txt">Table of n, a(n) for n = 1..10000</a>
%e A386207 6 is a term as 3^3 = 27 = 3 mod 6.
%e A386207 10 is a term as 5^5 = 3125 = 5 mod 10.
%o A386207 (Haskell) divisors n = [x | x <- [2..n], n `mod` x == 0]
%o A386207 property n = any (\x -> x^x `mod` n == x) $ divisors n
%o A386207 inRange t = [x | x <- [2..t], property x]
%o A386207 (PARI) isok(m) = fordiv(m, k, if((k>1) && (k<m) && (Mod(k, m)^k == k), return(1))); \\ _Michel Marcus_, Jul 16 2025
%Y A386207 Cf. A324455, A016825, A002997.
%K A386207 nonn
%O A386207 1,1
%A A386207 _Arjun Maneesh Agarwal_, Jul 15 2025