cp's OEIS Frontend

A052275 Number of distinct 6th powers mod n.

%I A052275 #30 Jan 27 2019 17:26:24
%S A052275 1,2,2,2,3,4,2,2,2,6,6,4,3,4,6,3,9,4,4,6,4,12,12,4,11,6,4,4,15,12,6,5,
%T A052275 12,18,6,4,7,8,6,6,21,8,8,12,6,24,24,6,8,22,18,6,27,8,18,4,8,30,30,12,
%U A052275 11,12,4,9,9,24,12,18,24,12,36,4,13,14,22,8,12,12
%N A052275 Number of distinct 6th powers mod n.
%C A052275 This sequence is multiplicative [Li]. - Leon P Smith, Apr 16 2005
%C A052275 Same as the number of distinct elements that are both squares and cubes mod n. - _Steven Finch_, Mar 01 2006
%H A052275 T. D. Noe, <a href="/A052275/b052275.txt">Table of n, a(n) for n = 1..1000</a>
%H A052275 S. R. Finch and Pascal Sebah, <a href="http://arXiv.org/abs/math.NT/0604465">Squares and Cubes Modulo n</a> (arXiv:math.NT/0604465).
%H A052275 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 A052275 Conjecture: a(2^n) = 1,2,2,2,3,5,9,18,... with g.f. ( 1-2*x^2-2*x^3-x^4-x^5-2*x^6 ) / ( (x-1)*(2*x-1)*(1+x)*(1+x+x^2)*(x^2-x+1) ). - _R. J. Mathar_, Sep 28 2017
%F A052275 Conjecture: a(3^n) = 1,2,2,4,10,28,82,.... with g.f. ( 1-x-4*x^2-2*x^3-2*x^4-2*x^5-3*x^6 ) / ( (x-1)*(3*x-1)*(1+x)*(x^2-x+1)*(1+x+x^2) ). - _R. J. Mathar_, Sep 28 2017
%p A052275 A052275 := proc(m)
%p A052275     {seq( modp(b^6,m),b=0..m-1) };
%p A052275     nops(%) ;
%p A052275 end proc:
%p A052275 seq(A052275(m),m=1..100) ; # _R. J. Mathar_, Sep 22 2017
%t A052275 Length[Union[#]]&/@Table[PowerMod[k,6,n],{n,100},{k,n}] (* _Zak Seidov_, Feb 17 2013 *)
%o A052275 (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^6%k), , 8)) \\ _Charles R Greathouse IV_, Sep 05 2013
%Y A052275 Cf. A000224 (squares), A046530 (cubic residues), A052273 (4th powers), A052274 (5th powers), A085310 (7th powers), A085311 (8th powers), A085312 (9th powers), A085313 (10th powers), A085314 (11th powers), A228849 (12th powers).
%K A052275 nonn,mult
%O A052275 1,2
%A A052275 _N. J. A. Sloane_, Feb 05 2000