A052275 Number of distinct 6th powers mod n.

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, 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, 11, 12, 4, 9, 9, 24, 12, 18, 24, 12, 36, 4, 13, 14, 22, 8, 12, 12

Offset: 1

Author

N. J. A. Sloane, Feb 05 2000

Keywords

• nonn
• mult

Comments

This sequence is multiplicative [Li]. - Leon P Smith, Apr 16 2005

Same as the number of distinct elements that are both squares and cubes mod n. - Steven Finch, Mar 01 2006

Links

• T. D. Noe, Table of n, a(n) for n = 1..1000
• S. R. Finch and Pascal Sebah, Squares and Cubes Modulo n (arXiv:math.NT/0604465).
• S. 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

Crossrefs

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).

Programs

Maple

A052275 := proc(m)
    {seq( modp(b^6,m),b=0..m-1) };
    nops(%) ;
end proc:
seq(A052275(m),m=1..100) ; # R. J. Mathar, Sep 22 2017

Mathematica

Length[Union[#]]&/@Table[PowerMod[k,6,n],{n,100},{k,n}] (* Zak Seidov, Feb 17 2013 *)

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

Formula

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

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