A380531 a(n) is the multiplicative order of -4 modulo prime(n); a(1) = 0 for completion.

0, 2, 1, 6, 10, 3, 4, 18, 22, 7, 10, 9, 5, 14, 46, 13, 58, 15, 66, 70, 18, 78, 82, 22, 24, 25, 102, 106, 9, 7, 14, 130, 17, 138, 37, 30, 13, 162, 166, 43, 178, 45, 190, 48, 49, 198, 210, 74, 226, 19, 58, 238, 12, 50, 8, 262, 67, 270, 23, 70

Offset: 1

Author

Jianing Song, Jun 27 2025

Keywords

• nonn
• easy

Comments

a(n) divides (p-1)/4 if p = prime(n) == 1 (mod 4), since (-4)^((p-1)/4) == (+-1+-i)^(p-1) == 1 (mod p), where i^2 == -1 (mod p).

Links

• Jianing Song, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A105876 (primes having primitive root -4).

Cf. bases 2..10: A014664, A062117, A082654, A211241, A211242, A211243, A211244, A211245, A002371.

Cf. bases -2..-10: A337878 (if first term 1), A380482, this sequence, A380532, A380533, A380540, A380541, A380542, A385222.

Programs

Mathematica

A380531[n_] := If[n == 1, 0, MultiplicativeOrder[-4, Prime[n]]];
Array[A380531, 100] (* Paolo Xausa, Jun 29 2025 *)

PARI

a(n,{k=-4}) = my(p = prime(n)); if(k%p==0, 0, znorder(Mod(k,p)))