A247257 - OEIS

%I #49 Dec 13 2024 11:24:27

%S 1,1,2,2,4,2,2,4,2,4,2,4,4,2,8,8,8,2,2,8,4,2,2,8,4,4,2,4,4,8,2,16,4,8,

%T 8,4,4,2,8,16,8,4,2,4,8,2,2,16,2,4,16,8,4,2,8,8,4,4,2,16,4,2,4,16,16,

%U 4,2,16,4,8,2,8,8,4,8,4,4,8,2,32

%N The number of octic characters modulo n.

%C Number of solutions to x^8 == 1 (mod n). - _Jianing Song_, Nov 10 2019

%H Charles R Greathouse IV, <a href="/A247257/b247257.txt">Table of n, a(n) for n = 1..10000</a>

%H Steven Finch, <a href="http://arxiv.org/abs/0907.4894">Quartic and octic characters modulo n</a>, arXiv:0907.4894 [math.NT], 2009.

%F Multiplicative with a(p^e) = p^min(e-1, 4) if p = 2, gcd(8, p-1) if p > 2. - _Jianing Song_, Nov 10 2019

%p A247257 := proc(n)

%p     local a,pf,p,r;

%p     a := 1 ;

%p     for pf in ifactors(n)[2] do

%p         p := op(1,pf);

%p         r := op(2,pf);

%p         if p = 2 then

%p             if r >= 5 then

%p                 a := a*16 ;

%p             else

%p                 a := a*op(r,[1,2,4,8]) ;

%p             end if;

%p         elif modp(p,4) = 3 then

%p             a := a*2;

%p         elif modp(p,8) = 5 then

%p             a := a*4;

%p         elif modp(p,8) = 1 then

%p             a := a*8;

%p         else

%p             error

%p         end if;

%p     end do:

%p     a ;

%p end proc:

%t g[p_, e_] := Which[p==2, 2^Min[e-1, 4], Mod[p, 4]==3, 2, Mod[p, 8]==5, 4, True, 8];

%t a[1] = 1; a[n_] := Times @@ g @@@ FactorInteger[n];

%t Array[a, 80] (* _Jean-François Alcover_, Nov 26 2017, after _Charles R Greathouse IV_ *)

%o (PARI) g(p,e)=if(p==2, 2^min(e-1,4), if(p%4==3, 2, if(p%8==5, 4, 8)))

%o a(n)=my(f=factor(n)); prod(i=1,#f~, g(f[i,1],f[i,2])) \\ _Charles R Greathouse IV_, Mar 02 2015

%Y Number of solutions to x^k == 1 (mod n): A060594 (k=2), A060839 (k=3), A073103 (k=4), A319099 (k=5), A319100 (k=6), A319101 (k=7), this sequence (k=8).

%K mult,nonn,easy

%O 1,3

%A _R. J. Mathar_, Mar 02 2015