A302254 Exponent of the group of the Gaussian integers in a reduced system modulo (1+i)^n.

1, 1, 2, 4, 4, 4, 4, 4, 8, 8, 16, 16, 32, 32, 64, 64, 128, 128, 256, 256, 512, 512, 1024, 1024, 2048, 2048, 4096, 4096, 8192, 8192, 16384, 16384, 32768, 32768, 65536, 65536, 131072, 131072, 262144, 262144, 524288, 524288, 1048576, 1048576, 2097152, 2097152, 4194304, 4194304, 8388608, 8388608

Offset

0,3

Comments

For n > 0, the number of elements in the group of the Gaussian integers in a reduced system modulo (1+i)^n is 2^(n-1).

Links

Andrew Howroyd, Table of n, a(n) for n = 0..1000

Index entries for linear recurrences with constant coefficients, signature (0,2).

Formula

For n > 5, a(n) = 2^(floor(n/2) - 1).

For even n, a(n) = A227334(2^(n/2)).

G.f.: (1 + x + 2*x^3 - 4*x^5 - 4*x^6 - 4*x^7)/(1 - 2*x^2). - Andrew Howroyd, Nov 06 2025

From Amiram Eldar, Apr 03 2026: (Start)

Sum_{n>=0} 1/a(n) = 17/4.

Sum_{n>=0} (-1)^n/a(n) = 1/4. (End)

Example

For Gaussian integer x such that (x, 1+i) = 1, x^4 - 1 = (x + 1)(x - 1)(x + i)(x - i) provides at least 7 factors of 1+i in total (and exactly 7 when x = 2+i), so a(7) = 4.

Mathematica

Join[{1, 1, 2, 4, 4, 4}, Table[2^(Floor[n/2] - 1), {n, 6, 50}]] (* Vincenzo Librandi, Apr 04 2018 *)

Prog

(Magma) [1, 1, 2, 4, 4, 4] cat [2^(Floor(n div 2)-1): n in [6..50]]; // Vincenzo Librandi, Apr 04 2018

Crossrefs

Cf. A016116, A079458, A227334.

Sequence in context: A260085 A159461 A046930 * A160409 A035645 A063440

Adjacent sequences: A302251 A302252 A302253 * A302255 A302256 A302257

Keyword

nonn,easy

Author

Jianing Song, Apr 04 2018

Status

approved