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A293482 The number of 5th powers in the multiplicative group modulo n. 6
1, 1, 2, 2, 4, 2, 6, 4, 6, 4, 2, 4, 12, 6, 8, 8, 16, 6, 18, 8, 12, 2, 22, 8, 4, 12, 18, 12, 28, 8, 6, 16, 4, 16, 24, 12, 36, 18, 24, 16, 8, 12, 42, 4, 24, 22, 46, 16, 42, 4, 32, 24, 52, 18, 8, 24, 36, 28, 58, 16, 12, 6, 36, 32, 48, 4, 66, 32, 44, 24, 14, 24, 72, 36, 8, 36, 12, 24, 78, 32, 54, 8, 82, 24 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,3

COMMENTS

The size of the set of numbers j^5 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..7548

R. J. Mathar, Size of the set of residues of integer powers of fixed exponent, (2017).

FORMULA

Conjecture: a(2^e) = 1 for e <= 1; a(2^e) = 2^(e-1) for e >= 1; a(5)=4; a(5^e) = 4*5^(e-2) for e > 1; a(p^e) = (p-1)*p^(e-1) for p == {2,3,4} (mod 5); a(p^e) = (p-1)*p^(e-1)/5 for p == 1 (mod 5). - R. J. Mathar, Oct 13 2017

a(n) = A000010(n)/A319099(n). This implies that the conjecture above is true. - Jianing Song, Nov 10 2019

MAPLE

A293482 := proc(n)

    local r, j;

    r := {} ;

    for j from 1 to n do

        if igcd(j, n)= 1 then

            r := r union { modp(j &^ 5, n) } ;

        end if;

    end do:

    nops(r) ;

end proc:

seq(A293482(n), n=1..120) ;

CROSSREFS

The number of k-th powers in the multiplicative group modulo n: A046073 (k=2), A087692 (k=3), A250207 (k=4), this sequence (k=5), A293483 (k=6), A293484 (k=7), A293485 (k=8).

Cf. A052274, A319099, A000010.

Sequence in context: A262550 A077651 A004085 * A086296 A096504 A277906

Adjacent sequences:  A293479 A293480 A293481 * A293483 A293484 A293485

KEYWORD

nonn,mult

AUTHOR

R. J. Mathar, Oct 10 2017

STATUS

approved

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Last modified November 30 15:14 EST 2020. Contains 338807 sequences. (Running on oeis4.)