A333433 a(n) is the n-th number m that divides n^m - 1 (or 0 if m does not exist).

1, 0, 4, 21, 8, 1555, 9, 6223, 40, 999, 20, 130801, 24, 4484077, 128, 117, 60, 118285781329, 42, 1432001198261, 104, 819, 72, 302508121, 81, 75625, 200, 61731, 78, 14507145975869, 72, 21958351241, 820, 12321, 289, 4375, 144

Offset: 1

Author

Seiichi Manyama, Mar 21 2020

Keywords

• nonn
• more

Comments

From Jinyuan Wang, Mar 25 2020: (Start)

For n > 2, n < a(n) < q^(n-1), where q is a prime factor of n - 1.

If p is a prime, then a(p^e+1) is divisible by p. Proof: we can prove that p | m for m > 1 and n = p^e + 1. If n^m == 1 (mod m) and m > 1 is the minimum value that cannot be divisible by p, then gcd(m, eulerphi(m)) = 1. Thus, m must be of the form q*p_2*...*p_k, where q < p_2 < ... < p_k. Note that q | (n^m - 1) = (n^q - 1)*(Sum_{i=0..(m/q)-1} n)^(i*q)) and n^q - 1 can never be divisible by q. Therefore, Sum_{i=0..(m/q)-1} n^(i*q) == n^(m/q) - 1 == 0 (mod q). Because n^(q-1) == 1 (mod q) and gcd(m/q, q - 1) = 1, then n == 1 (mod q), a contradiction! (End)

a(38) <= 14948925257859919. - Giovanni Resta, Apr 15 2020

Links

• OEIS Wiki, 2^n mod n

Crossrefs

Main diagonal of A333432.

Cf. A014960, A128358, A128360, A333430.

Programs

PARI

{a(n) = if(n==2, 0, my(cnt=0, k=0); while(cnt
				

Formula

a(n) = A333432(n,n).

Extensions

a(30)-a(37) from Giovanni Resta, Apr 15 2020