A385164 Let p = A002145(n) be the n-th prime == 3 (mod 4); 8*a(n) is the multiplicative order of 1+-i modulo p in Gaussian integers.

1, 3, 5, 9, 11, 5, 7, 23, 29, 33, 35, 39, 41, 51, 53, 7, 65, 69, 15, 81, 83, 89, 95, 99, 105, 37, 113, 119, 25, 131, 135, 47, 51, 155, 15, 173, 179, 183, 189, 191, 209, 43, 73, 221, 231, 233, 239, 243, 245, 83, 251, 261, 273, 281, 57, 293, 299, 303, 309, 45, 107, 323, 329, 11, 115

Offset

1,2

Links

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

Formula

a(n) = ord(-4,p)/2, where ord(a,p) is the multiplicative order of a modulo p.

Example

For A002145(4) = 19: Since (1+i)^(4k) = (-4)^k, we have (1+i)^72 == 1 (mod 19), and 72 is the smallest such exponent. Hence a(4) = 72/8 = 9.

Mathematica

MultiplicativeOrder[-4, #]/2 & /@ Select[Prime[Range[150]], Mod[#, 4] == 3 &] (* Amiram Eldar, Mar 28 2026 *)

Prog

(PARI) forprime(p=3, 1e3, if(p%4==3, print1(znorder(Mod(-4, p))/2, ", ")))

Crossrefs

Cf. A002145, A385163.

Sequence in context: A066769 A323765 A385227 * A270780 A386268 A304251

Adjacent sequences: A385161 A385162 A385163 * A385165 A385166 A385167

Keyword

nonn,easy

Author

Jianing Song, Jun 20 2025

Status

approved