Year-end appeal: Please make a donation to the OEIS Foundation to support ongoing development and maintenance of the OEIS. We are now in our 61st year, we have over 378,000 sequences, and we’ve reached 11,000 citations (which often say “discovered thanks to the OEIS”).
%I #31 Dec 01 2022 11:00:31
%S 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,
%T 3,14,29,120,31,8,33,136,35,6,37,152,39,40,41,168,43,22,15,184,47,6,7,
%U 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
%N Period of Kronecker symbol modulo n.
%C From _Jianing Song_, Nov 24 2018: (Start)
%C The sequence {Kronecker(k,n)} forms a Dirichlet character modulo n if and only if n !== 2 (mod 4).
%C 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)
%H Jianing Song, <a href="/A302138/b302138.txt">Table of n, a(n) for n = 1..10000</a>
%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Kronecker_symbol">Kronecker symbol</a>.
%F Multiplicative with a(p^e) = p, p > 2; a(2^e) = 2 for even e and 8 for odd e.
%F a(n) = A007947(n) if A007814(n) is even, else 4*A007947(n).
%F 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
%e The Kronecker symbol modulo 2 is 1, 0, -1, 0, -1, 0, 1, 0 with period 8, so a(2) = 8.
%e The Kronecker symbol modulo 9 is 1, 1, 0 with period 3, so a(9) = 3.
%t Array[Apply[Times, FactorInteger[#] /. {p_, e_} /; p > 0 :> If[p == 2, 2 + 6 Boole[OddQ@ e], p]] &, 100] (* _Michael De Vlieger_, Nov 25 2018 *)
%o (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
%Y Cf. A007947, A065463.
%Y Cf. A117888 (period of Kronecker(n,k)), A117889 (period of Kronecker(-n,k)).
%K nonn,easy,mult
%O 1,2
%A _Jianing Song_, Apr 02 2018