A293483 The number of 6th powers in the multiplicative group modulo n.

1, 1, 1, 1, 2, 1, 1, 1, 1, 2, 5, 1, 2, 1, 2, 2, 8, 1, 3, 2, 1, 5, 11, 1, 10, 2, 3, 1, 14, 2, 5, 4, 5, 8, 2, 1, 6, 3, 2, 2, 20, 1, 7, 5, 2, 11, 23, 2, 7, 10, 8, 2, 26, 3, 10, 1, 3, 14, 29, 2, 10, 5, 1, 8, 4, 5, 11, 8, 11, 2, 35, 1, 12, 6, 10, 3, 5, 2, 13, 4, 9, 20

Offset

1,5

Comments

The size of the set of numbers j^6 mod n, gcd(j,n)=1, 1 <= j <= n.

A000010(n) / a(n) is another multiplicative integer sequence.

Links

R. J. Mathar, Table of n, a(n) for n = 1..10132

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

Formula

Conjecture: a(2^e) = 1 for e <= 3; a(2^e) = 2^(e-3) for e >= 3; a(3^e) = 1 for e <= 2; a(3^e) = 3^(e-2) for e >= 2; a(p^e) = (p-1)*p^(e-1)/2 for p == 5 (mod 6); a(p^e) = (p-1)*p^(e-1)/6 for p == 1 (mod 6). - R. J. Mathar, Oct 13 2017

a(n) = A000010(n)/A319100(n). This implies that the conjecture above is true. - Jianing Song, Nov 10 2019

Maple

A293483 := proc(n)
    local r, j;
    r := {} ;
    for j from 1 to n do
        if igcd(j, n)= 1 then
            r := r union { modp(j &^ 6, n) } ;
        end if;
    end do:
    nops(r) ;
end proc:
seq(A293483(n), n=1..120) ;

Mathematica

a[n_] := EulerPhi[n]/Count[Range[0, n - 1]^6 - 1, k_ /; Divisible[k, n]];
Array[a, 100] (* Jean-François Alcover, May 24 2023 *)
f[p_, e_] := (p - 1)*p^(e - 1)/If[Mod[p, 6] == 1, 6, 2]; f[2, e_] := If[e <= 3, 1, 2^(e - 3)]; f[3, e_] := If[e <= 2, 1, 3^(e - 2)]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Aug 10 2023 *)

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), this sequence (k=6), A293484 (k=7), A293485 (k=8).

Cf. A052275, A319100, A000010.

Sequence in context: A106498 A093466 A257463 * A125761 A154950 A260089

Adjacent sequences: A293480 A293481 A293482 * A293484 A293485 A293486

Keyword

nonn,mult

Author

R. J. Mathar, Oct 10 2017

Status

approved