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A293482 The number of 5th powers in the multiplicative group modulo n. 5

%I

%S 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,

%T 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,

%U 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

%N The number of 5th powers in the multiplicative group modulo n.

%C The size of the set of numbers j^5 mod n, gcd(j,n)=1, 1 <= j <= n.

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

%H R. J. Mathar, <a href="/A293482/b293482.txt">Table of n, a(n) for n = 1..7548</a>

%H R. J. Mathar, <a href="http://www.mpia.de/~mathar/public/mathar20171110.pdf">Size of the set of residues of integer powers of fixed exponent</a>, (2017).

%F 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

%F a(n) = A000010(n)/A319099(n). This implies that the conjecture above is true. - _Jianing Song_, Nov 10 2019

%p A293482 := proc(n)

%p local r,j;

%p r := {} ;

%p for j from 1 to n do

%p if igcd(j,n)= 1 then

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

%p end if;

%p end do:

%p nops(r) ;

%p end proc:

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

%Y 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).

%Y Cf. A052274, A319099, A000010.

%K nonn,mult,changed

%O 1,3

%A _R. J. Mathar_, Oct 10 2017

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Last modified November 13 04:20 EST 2019. Contains 329085 sequences. (Running on oeis4.)