cp's OEIS Frontend

A052274 Number of distinct 5th powers mod n.

%I A052274 #25 Oct 22 2017 09:13:26
%S A052274 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 A052274 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 A052274 39,53,38,15,35,57,58,59,45,13,14,49,34,65,18,67,51,69,70
%N A052274 Number of distinct 5th powers mod n.
%C A052274 This sequence is multiplicative. - Leon P Smith, Apr 16 2005
%H A052274 T. D. Noe, <a href="/A052274/b052274.txt">Table of n, a(n) for n = 1..1000</a>
%H A052274 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 A052274 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 A052274 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 A052274 := proc(m)
%p A052274     {seq( modp(b^5,m),b=0..m-1) };
%p A052274     nops(%) ;
%p A052274 end proc:
%p A052274 seq(A052274(m),m=1..100) ; # _R. J. Mathar_, Sep 22 2017
%t A052274 With[{nn=100},Table[Length[Union[PowerMod[Range[nn],5,n]]],{n,nn}]] (* _Harvey P. Dale_, Mar 19 2016 *)
%o A052274 (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 A052274 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 A052274 nonn,mult
%O A052274 1,2
%A A052274 _N. J. A. Sloane_, Feb 05 2000