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A302138 Period of Kronecker symbol modulo n. 3
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 (list; graph; refs; listen; history; text; internal format)
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,changed

AUTHOR

Jianing Song, Apr 02 2018

STATUS

approved

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Last modified December 4 04:59 EST 2022. Contains 358544 sequences. (Running on oeis4.)