A302138 Period of Kronecker symbol modulo n.

1, 8, 3, 2, 5, 24, 7, 8, 3, 40, 11, 6, 13, 56, 15, 2, 17, 24, 19, 10, 21, 88, 23, 24, 5, 104, 3, 14, 29, 120, 31, 8, 33, 136, 35, 6, 37, 152, 39, 40, 41, 168, 43, 22, 15, 184, 47, 6, 7, 40, 51, 26, 53, 24, 55, 56, 57, 232, 59, 30, 61, 248, 21, 2, 65, 264, 67, 34, 69, 280, 71, 24, 73, 296, 15, 38, 77, 312, 79, 10, 3, 328, 83, 42, 85, 344, 87, 88, 89, 120, 91, 46, 93, 376, 95, 24, 97, 56, 33, 10

Offset

1,2

Comments

From Jianing Song, Nov 24 2018: (Start)

The sequence {Kronecker(k,n)} forms a Dirichlet character modulo n if and only if n !== 2 (mod 4).

Let n = 2^t*s, s odd, then a(n) = A117888(n) if and only if t is odd or s == 1 (mod 4) (or both); a(n) = A117889(n) if and only if t is odd or s == 3 (mod 4) (or both). (End)

Links

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

Wikipedia, Kronecker symbol.

Formula

Multiplicative with a(p^e) = p, p > 2; a(2^e) = 2 for even e and 8 for odd e.

a(n) = A007947(n) if A007814(n) is even, else 4*A007947(n).

Sum_{k=1..n} a(k) ~ c * n^2, where c = (49/50) * Product_{p prime} (1 - 1/(p*(p+1))) = (49/50) * A065463 = 0.690353... . - Amiram Eldar, Dec 01 2022

Example

The Kronecker symbol modulo 2 is 1, 0, -1, 0, -1, 0, 1, 0 with period 8, so a(2) = 8.
The Kronecker symbol modulo 9 is 1, 1, 0 with period 3, so a(9) = 3.

Mathematica

Array[Apply[Times, FactorInteger[#] /. {p_, e_} /; p > 0 :> If[p == 2, 2 + 6 Boole[OddQ@ e], p]] &, 100] (* Michael De Vlieger, Nov 25 2018 *)

Prog

(PARI) a(n)={my(f=factor(n)); prod(i=1, #f~, if(f[i, 1]==2 && f[i, 2]%2, 8, f[i, 1]))} \\ Andrew Howroyd, Apr 29 2018

Crossrefs

Cf. A007947, A065463.

Cf. A117888 (period of Kronecker(n,k)), A117889 (period of Kronecker(-n,k)).

Sequence in context: A346713 A154014 A063568 * A198494 A100668 A194159

Adjacent sequences: A302135 A302136 A302137 * A302139 A302140 A302141

Keyword

nonn,easy,mult

Author

Jianing Song, Apr 02 2018

Status

approved