A046530 Number of distinct cubic residues mod n.

1, 2, 3, 3, 5, 6, 3, 5, 3, 10, 11, 9, 5, 6, 15, 10, 17, 6, 7, 15, 9, 22, 23, 15, 21, 10, 7, 9, 29, 30, 11, 19, 33, 34, 15, 9, 13, 14, 15, 25, 41, 18, 15, 33, 15, 46, 47, 30, 15, 42, 51, 15, 53, 14, 55, 15, 21, 58, 59, 45, 21, 22, 9, 37, 25, 66, 23, 51, 69, 30, 71, 15, 25, 26, 63

Offset: 1

Author

David W. Wilson

Keywords

• nonn
• mult
• easy
• nice

Comments

In other words, number of distinct cubes mod n. - N. J. A. Sloane, Oct 05 2024

Cubic analog of A000224. - Steven Finch, Mar 01 2006

A074243 contains values of n such that a(n) = n. - Dmitri Kamenetsky, Nov 03 2012

Links

• T. D. Noe, Table of n, a(n) for n = 1..1000
• Steven R. Finch and Pascal Sebah, Squares and Cubes Modulo n, arXiv:math/0604465 [math.NT], 2005-2016.
• Shuguang Li, On the number of elements with maximal order in the multiplicative group modulo n, Acta Arithm. 86 (2) (1998) 113, see proof of theorem 2.1.
• Param Parekh, Paavan Parekh, Sourav Deb, and Manish K. Gupta, On the Classification of Weierstrass Elliptic Curves over Z_n, arXiv:2310.11768 [cs.CR], 2023. See p. 7.

Crossrefs

For number of k-th power residues mod n, see A000224 (k=2), A052273 (k=4), A052274 (k=5), A052275 (k=6), A085310 (k=7), A085311 (k=8), A085312 (k=9), A085313 (k=10), A085314 (k=12), A228849 (k=13).

Cf. A000578, A087786, A257301.

Programs

Haskell

import Data.List (nub)
a046530 n = length $ nub $ map (`mod` n) $
                           take (fromInteger n) $ tail a000578_list
-- Reinhard Zumkeller, Aug 01 2012

Maple

A046530 := proc(n)
        local a,pf ;
        a := 1 ;
        if n = 1 then
                return 1;
        end if;
        for i in  ifactors(n)[2] do
                p := op(1,i) ;
                e := op(2,i) ;
                if p = 3 then
                        if e mod 3 = 0 then
                                a := a*(3^(e+1)+10)/13 ;
                        elif e mod 3 = 1 then
                                a := a*(3^(e+1)+30)/13 ;
                        else
                                a := a*(3^(e+1)+12)/13 ;
                        end if;
                elif p mod 3 = 2 then
                        if e mod 3 = 0 then
                                a := a*(p^(e+2)+p+1)/(p^2+p+1) ;
                        elif e mod 3 = 1 then
                                a := a*(p^(e+2)+p^2+p)/(p^2+p+1) ;
                        else
                                a := a*(p^(e+2)+p^2+1)/(p^2+p+1) ;
                        end if;
                else
                        if e mod 3 = 0 then
                                a := a*(p^(e+2)+2*p^2+3*p+3)/3/(p^2+p+1) ;
                        elif e mod 3 = 1 then
                                a := a*(p^(e+2)+3*p^2+3*p+2)/3/(p^2+p+1) ;
                        else
                                a := a*(p^(e+2)+3*p^2+2*p+3)/3/(p^2+p+1) ;
                        end if;
                end if;
        end do:
        a ;
end proc:
seq(A046530(n),n=1..40) ; # R. J. Mathar, Nov 01 2011

Mathematica

Length[Union[#]]& /@ Table[Mod[k^3, n], {n, 75}, {k, n}] (* Jean-François Alcover, Aug 30 2011 *)
Length[Union[#]]&/@Table[PowerMod[k,3,n],{n,80},{k,n}] (* Harvey P. Dale, Aug 12 2015 *)

PARI

g(p,e)=if(p==3,(3^(e+1)+if(e%3==1,30,if(e%3,12,10)))/13, if(p%3==2, (p^(e+2)+if(e%3==1,p^2+p,if(e%3,p^2+1,p+1)))/(p^2+p+1),(p^(e+2)+if(e%3==1,3*p^2+3*p+2, if(e%3,3*p^2+2*p+3,2*p^2+3*p+3)))/3/(p^2+p+1)))
a(n)=my(f=factor(n));prod(i=1,#f[,1],g(f[i,1],f[i,2])) \\ Charles R Greathouse IV, Jan 03 2013

PARI

a(n)={my(f=factor(n)); prod(i=1, #f~, my(p=f[i,1], e=f[i,2]); 1 + sum(i=0, (e-1)\3, if(p%3==1 || (p==3&&3*iAndrew Howroyd, Jul 17 2018

Formula

a(n) = n - A257301(n). - Stanislav Sykora, Apr 21 2015

a(2^n) = A046630(n). a(3^n) = A046631(n). a(5^n) = A046633(n). a(7^n) = A046635(n). - R. J. Mathar, Sep 28 2017

Multiplicative with a(p^e) = 1 + Sum_{i=0..floor((e-1)/3)} (p - 1)*p^(e-3*i-1)/k where k = 3 if (p = 3 and 3*i + 1 = e) or (p mod 3 = 1) otherwise k = 1. - Andrew Howroyd, Jul 17 2018

Sum_{k=1..n} a(k) ~ c * n^2/log(n)^(1/3), where c = (6/(13*Gamma(2/3))) * (2/3)^(-1/3) * Product_{p prime == 2 (mod 3)} (1 - (p^2+1)/((p^2+p+1)*(p^2-p+1)*(p+1))) * (1-1/p)^(-1/3) * Product_{p prime == 1 (mod 3)} (1 - (2*p^4+3*p^2+3)/(3*(p^2+p+1)*(p^2-p+1)*(p+1))) * (1-1/p)^(-1/3) = 0.48487418844474389597... (Finch and Sebah, 2006). - Amiram Eldar, Oct 18 2022