A293485 The number of 8th powers in the multiplicative group modulo n.

1, 1, 1, 1, 1, 1, 3, 1, 3, 1, 5, 1, 3, 3, 1, 1, 2, 3, 9, 1, 3, 5, 11, 1, 5, 3, 9, 3, 7, 1, 15, 1, 5, 2, 3, 3, 9, 9, 3, 1, 5, 3, 21, 5, 3, 11, 23, 1, 21, 5, 2, 3, 13, 9, 5, 3, 9, 7, 29, 1, 15, 15, 9, 2, 3, 5, 33, 2, 11, 3, 35, 3, 9, 9, 5, 9, 15, 3, 39, 1, 27, 5, 41, 3, 2

Offset

1,7

Comments

The size of the set of numbers j^8 mod n, gcd(j,n)=1, 1<=j<=n.

Links

Antti Karttunen, Table of n, a(n) for n = 1..16384

Richard J. Mathar, Size of the Set of Residues of Integer Powers of Fixed Exponent, research paper, 2017.

Formula

A000010(n) / a(n) = A247257(n).

Multiplicative with a(2^e) = 1 for e<=4, a(2^e) = 2^(e-5) for e>=5; a(p^e) = (p-1)*p^(e-1)/8 for p == 1 (mod 8); a(p^e) = (p-1)*p^(e-1)/4 for p == 5 (mod 8); a(p^e) = (p-1)*p^(e-1)/2 for p == {3,7} (mod 8). - R. J. Mathar, Oct 15 2017 [corrected by Georg Fischer, Jul 21 2022]

Maple

A293485 := proc(n)
    local r, j;
    r := {} ;
    for j from 1 to n do
        if igcd(j, n)= 1 then
            r := r union { modp(j &^ 8, n) } ;
        end if;
    end do:
    nops(r) ;
end proc:
seq(A293485(n), n=1..120) ;

Mathematica

a[n_] := EulerPhi[n]/Count[Range[0, n - 1]^8 - 1, k_ /; Divisible[k, n]];
Array[a, 100] (* Jean-François Alcover, May 24 2023 *)
f[p_, e_] := (p - 1)*p^(e - 1)/Switch[Mod[p, 8], 1, 8, 5, 4, _, 2]; f[2, e_] := If[e <= 4, 1, 2^(e - 5)]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Aug 10 2023 *)

Prog

(PARI)
\\ The following two functions by Charles R Greathouse IV, from A247257:
g(p, e) = if(p==2, 2^min(e-1, 4), if(p%4==3, 2, if(p%8==5, 4, 8)));
A247257(n) = my(f=factor(n)); prod(i=1, #f~, g(f[i, 1], f[i, 2]));
A293485(n) = (eulerphi(n)/A247257(n)); \\ Antti Karttunen, Dec 05 2017

Crossrefs

The number of k-th powers in the multiplicative group modulo n: A046073 (k=2), A087692 (k=3), A250207 (k=4), A293482 (k=5), A293483 (k=6), A293484 (k=7), this sequence (k=8).

Cf. A085311, A247257 (order of the kernel isomorphism of Z/nZ to this group), A000010.

Sequence in context: A322821 A213621 A053575 * A250207 A216319 A355001

Adjacent sequences: A293482 A293483 A293484 * A293486 A293487 A293488

Keyword

nonn,mult

Author

R. J. Mathar, Oct 10 2017

Status

approved