A395286 - OEIS

%I #17 May 05 2026 00:16:44

%S 9,15,29,52

%N Positive integers m such that gcd(m^10 - 1, m! - 1) > 1.

%C For m = 9, gcd(m^10 - 1, m! - 1) = 121 = 11^2, and this shows that there exist integers m > 1 and k > 1 such that gcd(m^k - 1, m! - 1) is not squarefree.

%C No additional terms are known for m <= 100000 (it is not known whether the sequence is finite).

%C No additional terms for m <= 5*10^6. - _Michael S. Branicky_, May 01 2026

%e 15 and 29 are terms since gcd(15^10 - 1, 15! - 1) = gcd(29^10 - 1, 29! - 1) = 31.

%o (PARI) isok(m) = gcd(m^10 - 1, m! - 1) > 1; \\ _Michel Marcus_, Apr 20 2026

%o (Python)

%o import math

%o def isok(m): return math.gcd(m**10-1, math.factorial(m)-1) > 1  # _Jwalin Bhatt_, Apr 28 2026

%Y Cf. A033312, A069095.

%K nonn,hard,more

%O 1,1

%A _Marco Ripà_, Apr 18 2026