%I #25 Oct 22 2017 09:13:26

%S 1,2,3,3,5,6,7,5,7,10,3,9,13,14,15,9,17,14,19,15,21,6,23,15,5,26,19,

%T 21,29,30,7,17,9,34,35,21,37,38,39,25,9,42,43,9,35,46,47,27,43,10,51,

%U 39,53,38,15,35,57,58,59,45,13,14,49,34,65,18,67,51,69,70

%N Number of distinct 5th powers mod n.

%C This sequence is multiplicative. - Leon P Smith, Apr 16 2005

%H T. D. Noe, <a href="/A052274/b052274.txt">Table of n, a(n) for n = 1..1000</a>

%H S. Li, <a href="http://pldml.icm.edu.pl/pldml/element/bwmeta1.element.bwnjournal-article-aav86i2p113bwm">On the number of elements with maximal order in the multiplicative group modulo n</a>, Acta Arithm. 86 (2) (1998) 113, see proof of theorem 2.1

%F Conjecture: a(5^e) = 1+floor[(5-1)*5^(e+3)/(5^5-1)] if e == {0,2,3,4} (mod 5). a(5^e) = 5+floor[(5-1)*5^(e+3)/(5^5-1)] if e==1 (mod 5). - _R. J. Mathar_, Oct 22 2017

%F Conjecture: a(p^e) = 1+floor[(p-1)*p^(e+4)/{gcd(p-1,5)*(p^5-1)}] for primes p<>5 - _R. J. Mathar_, Oct 22 2017

%p A052274 := proc(m)

%p {seq( modp(b^5,m),b=0..m-1) };

%p nops(%) ;

%p end proc:

%p seq(A052274(m),m=1..100) ; # _R. J. Mathar_, Sep 22 2017

%t With[{nn=100},Table[Length[Union[PowerMod[Range[nn],5,n]]],{n,nn}]] (* _Harvey P. Dale_, Mar 19 2016 *)

%o (PARI) a(n)=my(f=factor(n)); prod(i=1, #f[, 1], my(k=f[i, 1]^f[i, 2]); #vecsort(vector(k, i, i^5%k), , 8)) \\ _Charles R Greathouse IV_, Sep 05 2013

%Y Cf. A000224 (squares), A046530 (cubic residues), A052273 (4th powers), A052275 (6th powers), A085310 (7th powers), A085311 (8th powers), A085312 (9th powers), A085313 (10th powers), A085314 (11th powers), A228849 (12th powers).

%K nonn,mult

%O 1,2

%A _N. J. A. Sloane_, Feb 05 2000