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A247257 The number of octic characters modulo n.

%I A247257 #49 Dec 13 2024 11:24:27
%S A247257 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 A247257 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 A247257 4,2,16,4,8,2,8,8,4,8,4,4,8,2,32
%N A247257 The number of octic characters modulo n.
%C A247257 Number of solutions to x^8 == 1 (mod n). - _Jianing Song_, Nov 10 2019
%H A247257 Charles R Greathouse IV, <a href="/A247257/b247257.txt">Table of n, a(n) for n = 1..10000</a>
%H A247257 Steven Finch, <a href="http://arxiv.org/abs/0907.4894">Quartic and octic characters modulo n</a>, arXiv:0907.4894 [math.NT], 2009.
%F A247257 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 A247257 := proc(n)
%p A247257     local a,pf,p,r;
%p A247257     a := 1 ;
%p A247257     for pf in ifactors(n)[2] do
%p A247257         p := op(1,pf);
%p A247257         r := op(2,pf);
%p A247257         if p = 2 then
%p A247257             if r >= 5 then
%p A247257                 a := a*16 ;
%p A247257             else
%p A247257                 a := a*op(r,[1,2,4,8]) ;
%p A247257             end if;
%p A247257         elif modp(p,4) = 3 then
%p A247257             a := a*2;
%p A247257         elif modp(p,8) = 5 then
%p A247257             a := a*4;
%p A247257         elif modp(p,8) = 1 then
%p A247257             a := a*8;
%p A247257         else
%p A247257             error
%p A247257         end if;
%p A247257     end do:
%p A247257     a ;
%p A247257 end proc:
%t A247257 g[p_, e_] := Which[p==2, 2^Min[e-1, 4], Mod[p, 4]==3, 2, Mod[p, 8]==5, 4, True, 8];
%t A247257 a[1] = 1; a[n_] := Times @@ g @@@ FactorInteger[n];
%t A247257 Array[a, 80] (* _Jean-François Alcover_, Nov 26 2017, after _Charles R Greathouse IV_ *)
%o A247257 (PARI) g(p,e)=if(p==2, 2^min(e-1,4), if(p%4==3, 2, if(p%8==5, 4, 8)))
%o A247257 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 A247257 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 A247257 mult,nonn,easy
%O A247257 1,3
%A A247257 _R. J. Mathar_, Mar 02 2015