A046530 -id:A046530 - cp's OEIS frontend

A123365 Values of k such that A046530(k) = (k+2)/3, where A046530(k) is the number of distinct residues of cubes mod k.

1, 7, 13, 19, 31, 37, 43, 61, 67, 73, 79, 97, 103, 109, 127, 139, 151, 157, 163, 181, 193, 199, 211, 223, 229, 241, 271, 277, 283, 307, 313, 331, 337, 349, 367, 373, 379, 397, 409, 421, 433, 439, 457, 463, 487, 499, 523, 541, 547, 571, 577, 601, 607, 613, 619

Offset: 1

John W. Layman, Oct 12 2006

nonn

Conjecture: With the exception of the first term a(1)=1, this is exactly the sequence of primes of the form 6k+1 (A002476). This has been verified up to a(n)=2000.

R. J. Mathar, Table of n, a(n) for n = 1..10000
Jon Maiga, Computer-generated formulas for A123365, Sequence Machine.

Cf. A002476, A046530, A123722, A123723, A177965.

n := 1 :
a := 1 :
while n <= 10000 do
    printf("%d %d\n",n,a) ;
    a := a+1 ;
    while A046530(a) <> (a+2)/3 do
        a := a+1 ;
    end do:
    n := n+1 ;
end do: # creates b-file, R. J. Mathar, Sep 21 2017

A330695 Indices k at which record low values of the ratio A046530(k)/k occur.

Original entry on oeis.org

1, 4, 7, 9, 27, 36, 63, 117, 171, 189, 252, 351, 468, 504, 756, 819, 1197, 1953, 2223, 2457, 3276, 3591, 4788, 5733, 6552, 9576, 9828, 14364, 15561, 25389, 30303, 35217, 37107, 44289, 46683, 62244, 76167, 90909, 101556, 108927, 124488, 186732, 296856, 304668

Offset: 1

Jon E. Schoenfield, Dec 26 2019

nonn

A046530(k) is the number of distinct cubic residues mod k.

			   n  a(n)=k   A046530(k)  ratio A046530(k)/k
  --  ------   ----------  ------------------
   1       1         1     1.0000000000000...
   2       4         3     0.7500000000000...
   3       7         3     0.4285714285714...
   4       9         3     0.3333333333333...
   5      27         7     0.2592592592592...
   6      36         9     0.2500000000000...
   7      63         9     0.1428571428571...
   8     117        15     0.1282051282051...
   9     171        21     0.1228070175438...
  10     189        21     0.1111111111111...
  11     252        27     0.1071428571428...
  12     351        35     0.0997150997150...
  13     468        45     0.0961538461538...
  14     504        45     0.0892857142857...
  15     756        63     0.0833333333333...
  16     819        45     0.0549450549450...
  17    1197        63     0.0526315789473...
  18    1953        99     0.0506912442396...
  19    2223       105     0.0472334682860...
  20    2457       105     0.0427350427350...
  21    3276       135     0.0412087912087...
  22    3591       147     0.0409356725146...
  23    4788       189     0.0394736842105...
  24    5733       225     0.0392464678178...
  25    6552       225     0.0343406593406...
  26    9576       315     0.0328947368421...
  27    9828       315     0.0320512820512...
  28   14364       441     0.0307017543859...
  29   15561       315     0.0202429149797...
  30   25389       495     0.0194966323998...
  31   30303       585     0.0193050193050...
  32   35217       675     0.0191668796319...
  33   37107       693     0.0186757215619...
  34   44289       819     0.0184921763869...
  35   46683       735     0.0157444894286...
  36   62244       945     0.0151821862348...
  37   76167      1155     0.0151640474221...
  38   90909      1365     0.0150150150150...
  39  101556      1485     0.0146224742998...
  40  108927      1575     0.0144592249855...
  41  124488      1575     0.0126518218623...
  42  186732      2205     0.0118083670715...
  43  296856      3465     0.0116723259762...
  44  304668      3465     0.0113730355665...

Cf. A046530.

A000224 Number of squares mod n.

Original entry on oeis.org

1, 2, 2, 2, 3, 4, 4, 3, 4, 6, 6, 4, 7, 8, 6, 4, 9, 8, 10, 6, 8, 12, 12, 6, 11, 14, 11, 8, 15, 12, 16, 7, 12, 18, 12, 8, 19, 20, 14, 9, 21, 16, 22, 12, 12, 24, 24, 8, 22, 22, 18, 14, 27, 22, 18, 12, 20, 30, 30, 12, 31, 32, 16, 12, 21, 24, 34, 18, 24, 24, 36, 12

Offset: 1

N. J. A. Sloane

For any n > 2, there are quadratic nonresidues mod n, so a(n) < n. - Charles R Greathouse IV, Oct 28 2022
Conjecture: n^2 == 1 (mod a(n)*(a(n)-1)) if and only if n is an odd prime. - Thomas Ordowski, Apr 13 2025
This conjecture holds at least up to n = 10^8. - Michel Marcus, Apr 13 2025

			The sequence of squares (A000290) modulo 10 reads 0, 1, 4, 9, 6, 5, 6, 9, 4, 1, 0, 1, 4, 9, 6, 5, 6, 9, 4, 1,... and this reduced sequence contains a(10) = 6 different values, {0,1,4,5,6,9}. - _R. J. Mathar_, Oct 10 2014

T. D. Noe, Table of n, a(n) for n = 1..10000
Imanuel Chen and Michael Z. Spivey, Integral Generalized Binomial Coefficients of Multiplicative Functions, Preprint 2015; Summer Research Paper 238, Univ. Puget.
Steven R. Finch and Pascal Sebah, Squares and Cubes Modulo n, arXiv:math/0604465 [math.NT], 2006-2016.
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. 6.
E. J. F. Primrose, The number of quadratic residues mod m, Math. Gaz. v. 61 (1977) n. 415, 60-61.
Walter D. Stangl, Counting Squares in Z_n, Math. Mag. 69 (1996) 285-289.

Cf. A095972, A046530 (cubic residues), A052273 (4th powers), A052274 (5th powers), A052275 (6th powers), A085310 (7th powers), A085311 (8th powers), A085312 (9th powers), A085313 (10th powers), A085314 (11th powers), A228849 (12th powers).

a000224 n = product $ zipWith f (a027748_row n) (a124010_row n) where
   f 2 e = 2 ^ e `div` 6 + 2
   f p e = p ^ (e + 1) `div` (2 * p + 2) + 1
-- Reinhard Zumkeller, Aug 01 2012

A000224 := proc(m)
    {seq( modp(b^2,m),b=0..m-1) };
    nops(%) ;
end proc: # Emeric Deutsch
# 2nd implementation
A000224 := proc(n)
    local a,ifs,f,p,e,c ;
    a := 1 ;
    ifs := ifactors(n)[2] ;
    for f in ifs do
        p := op(1,f) ;
        e := op(2,f) ;
        if p = 2 then
            if type(e,'odd') then
                a := a*(2^(e-1)+5)/3 ;
            else
                a := a*(2^(e-1)+4)/3 ;
            end if;
        else
            if type(e,'odd') then
                c := 2*p+1 ;
            else
                c := p+2 ;
            end if;
            a := a*(p^(e+1)+c)/2/(p+1) ;
        end if;
    end do:
    a ;
end proc: # R. J. Mathar, Oct 10 2014

Length[Union[#]]& /@ Table[Mod[k^2, n], {n, 65}, {k, n}] (* Jean-François Alcover, Aug 30 2011 *)
a[2] = 2; a[n_] := a[n] = Switch[fi = FactorInteger[n], {{, 1}}, (fi[[1, 1]] + 1)/2, {{2, }}, 3/2 + 2^fi[[1, 2]]/6 + (-1)^(fi[[1, 2]]+1)/6, {{, }}, {p, k} = fi[[1]]; 3/4 + (p-1)*(-1)^(k+1)/(4*(p+1)) + p^(k+1)/(2*(p+1)), , Times @@ Table[ a[Power @@ f], {f, fi}]]; Table[a[n], {n, 1, 100}] (* _Jean-François Alcover, Mar 09 2015 *)

a(n) = local(v,i); v = vector(n,i,0); for(i=0, floor(n/2),v[i^2%n+1] = 1); sum(i=1,n,v[i]) \\ Franklin T. Adams-Watters, Nov 05 2006

a(n)=my(f=factor(n));prod(i=1,#f[,1],if(f[i,1]==2,2^f[1,2]\6+2,f[i,1]^(f[i,2]+1)\(2*f[i,1]+2)+1)) \\ Charles R Greathouse IV, Jul 15 2011

from math import prod
from sympy import factorint
def A000224(n): return prod((p**(e+1)//((p+1)*(q:=1+(p==2)))>>1)+q for p, e in factorint(n).items()) # Chai Wah Wu, Oct 07 2024

a(n) = A105612(n) + 1.
Multiplicative with a(p^e) = floor(p^e/6) + 2 if p = 2; floor(p^(e+1)/(2p + 2)) + 1 if p > 2. - David W. Wilson, Aug 01 2001
a(2^n) = A023105(n). a(3^n) = A039300(n). a(5^n) = A039302(n). a(7^n) = A039304(n). - R. J. Mathar, Sep 28 2017
Sum_{k=1..n} a(k) ~ c * n^2/sqrt(log(n)), where c = (17/(32*sqrt(Pi))) * Product_{p prime} (1 - (p^2+2)/(2*(p^2+1)*(p+1))) * (1-1/p)^(-1/2) = 0.37672933209687137604... (Finch and Sebah, 2006). - Amiram Eldar, Oct 18 2022
If p is an odd prime, then a(p) = (p + 1)/2. - Thomas Ordowski, Apr 09 2025

A052273 Number of distinct 4th powers mod n.

Original entry on oeis.org

1, 2, 2, 2, 2, 4, 4, 2, 4, 4, 6, 4, 4, 8, 4, 2, 5, 8, 10, 4, 8, 12, 12, 4, 6, 8, 10, 8, 8, 8, 16, 4, 12, 10, 8, 8, 10, 20, 8, 4, 11, 16, 22, 12, 8, 24, 24, 4, 22, 12, 10, 8, 14, 20, 12, 8, 20, 16, 30, 8, 16, 32, 16, 6, 8, 24, 34, 10, 24, 16, 36, 8, 19, 20, 12, 20

Offset: 1

N. J. A. Sloane, Feb 05 2000

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

T. D. Noe, Table of n, a(n) for n = 1..1000
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
Samer Seraj, Counting general power residues, Notes on Number Theory and Discrete Mathematics, 28:4 (2022), 730-743. Substituting k=4 into Theorem 1.1 gives a closed formula.
Samer Seraj, Resolution of Mathar's Conjectures on Counting Power Residues, #A62 INTEGERS 23 (2023).

Cf. A000224 (squares), A046530 (cubic residues), A052274 (5th powers), A052275 (6th powers), A085310 (7th powers), A085311 (8th powers), A085312 (9th powers), A085313 (10th powers), A085314 (11th powers), A228849 (12th powers).

A052273 := proc(n,k) local i; nops({seq(i^k mod n,i=0..n-1)}); end; # number of k-th powers mod n

a[n_] := Table[PowerMod[i, 4, n], {i, 0, n-1}] // Union // Length;
Array[a, 100] (* Jean-François Alcover, Mar 24 2020 *)

a(n)=my(f=factor(n)); prod(i=1,#f[,1],my(k=f[i,1]^f[i,2]); #vecsort(vector(k,i,i^4%k),,8)) \\ Charles R Greathouse IV, May 26 2013

\\ general formula for k-th powers, see Seraj link
h(p,e,k=4)=my(a=(p-1)/gcd(k,p-1),b=if(k%2+p%2,0,valuation(k,p)+1)+p%2*valuation(k,p),g=(e-1)%k+1,G=p^g,B=p^(b+1),K=p^k,E=p^e); a*(K/B*(E-G)/(K-1)+ceil(G/B))+1
a(n,f=factor(n),k=4)=prod(i=1,#f~, h(f[i,1],f[i,2],k)) \\ Charles R Greathouse IV, Nov 09 2022

from math import prod
from sympy import factorint
def A052273(n): return prod(1+(p**e//15+bool(e&3) if p==2 else (p-1)*p**(e+3)//((4 if p&3==1 else 2)*(p**4-1))) for p, e in factorint(n).items()) # Chai Wah Wu, Apr 09 2025

Conjecture: a(2^e) = 1 + floor(2^e/(2^4-1)) if e == 0 (mod 4). a(2^e) = 2 + floor(2^e/(2^4-1)) if e == {1,2,3} mod 4. - R. J. Mathar, Oct 22 2017
Conjecture: a(p^e) = 1 + floor((p-1)*p^(e+3)/(gcd(p-1,4)*(p^4-1))) for odd primes p. - R. J. Mathar, Oct 22 2017
From Samer Seraj, Nov 09 2022: (Start)
The above conjectures are correct, and a unified form is:
a(p^m) = alpha*((p^3 / p^beta)*((p^m - p^gamma)/(p^4 -1)) + ceiling((p^gamma)/(p^(beta+1)))) + 1, where p is any prime, m is any positive integer, alpha = (p-1)/gcd(4,p-1), beta = 3 if p = 2 or beta = 0 if p is odd, and gamma = 4 if 4|m or gamma = (m mod 4) otherwise. (End)

A052274 Number of distinct 5th powers mod n.

Original entry on oeis.org

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, 21, 29, 30, 7, 17, 9, 34, 35, 21, 37, 38, 39, 25, 9, 42, 43, 9, 35, 46, 47, 27, 43, 10, 51, 39, 53, 38, 15, 35, 57, 58, 59, 45, 13, 14, 49, 34, 65, 18, 67, 51, 69, 70

Offset: 1

N. J. A. Sloane, Feb 05 2000

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

T. D. Noe, Table of n, a(n) for n = 1..1000
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

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).

A052274 := proc(m)
    {seq( modp(b^5,m),b=0..m-1) };
    nops(%) ;
end proc:
seq(A052274(m),m=1..100) ; # R. J. Mathar, Sep 22 2017

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

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

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

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

A052275 Number of distinct 6th powers mod n.

Original entry on oeis.org

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

N. J. A. Sloane, Feb 05 2000

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

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

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).

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

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

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

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

A085310 Number of distinct 7th powers modulo n.

Original entry on oeis.org

1, 2, 3, 3, 5, 6, 7, 5, 7, 10, 11, 9, 13, 14, 15, 9, 17, 14, 19, 15, 21, 22, 23, 15, 21, 26, 19, 21, 5, 30, 31, 17, 33, 34, 35, 21, 37, 38, 39, 25, 41, 42, 7, 33, 35, 46, 47, 27, 7, 42, 51, 39, 53, 38, 55, 35, 57, 10, 59, 45, 61, 62, 49, 33, 65, 66, 67, 51, 69, 70, 11, 35, 73, 74

Offset: 1

Labos Elemer, Jun 27 2003

Compare with enigmatic similarity of analogous odd-th power counts to A055653.

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

T. D. Noe, Table of n, a(n) for n = 1..1000
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

Cf. A000224[k=2], A046530[k=3], A052273[k=4], A052274[k=5], A052275[k=6], A085311[k=8], A085312[k=9], A085313[k=10], A085314[k=11], A228849[k=12], A055653.

A085310 := proc(m)
    {seq( modp(b^7,m),b=0..m-1) };
    nops(%) ;
end proc:
seq(A085310(m),m=1..100) ; # R. J. Mathar, Sep 22 2017

a[n_] := Table[PowerMod[i, 7, n], {i, 0, n - 1}] // Union // Length;
Array[a, 100] (* Jean-François Alcover, Mar 24 2020 *)

a(n)=my(f=factor(n)); prod(i=1, #f[, 1], my(k=f[i, 1]^f[i, 2]); #vecsort(vector(k, i, i^7%k), , 8)) \\ Charles R Greathouse IV, Sep 05 2013

A085311 Number of distinct 8th powers modulo n.

Original entry on oeis.org

1, 2, 2, 2, 2, 4, 4, 2, 4, 4, 6, 4, 4, 8, 4, 2, 3, 8, 10, 4, 8, 12, 12, 4, 6, 8, 10, 8, 8, 8, 16, 2, 12, 6, 8, 8, 10, 20, 8, 4, 6, 16, 22, 12, 8, 24, 24, 4, 22, 12, 6, 8, 14, 20, 12, 8, 20, 16, 30, 8, 16, 32, 16, 3, 8, 24, 34, 6, 24, 16, 36, 8, 10, 20, 12, 20, 24, 16, 40, 4, 28, 12, 42, 16, 6

Offset: 1

Labos Elemer, Jun 27 2003

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

T. D. Noe, Table of n, a(n) for n = 1..1000
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
R. J. Mathar, Size of the set of residues of integer powers of fixed exponent, (2017).

Cf. A000224[k=2], A046530[k=3], A052273[k=4], A052274[k=5], A052275[k=6], A085310[k=7], A085312[k=9], A085313[k=10], A085314[k=11], A228849[k=12], A055653.

A085311 := proc(m)
    {seq( modp(b^8,m),b=0..m-1) };
    nops(%) ;
end proc:
seq(A085311(m),m=1..100) ; # R. J. Mathar, Sep 22 2017

a[n_] := Table[PowerMod[i, 8, n], {i, 0, n - 1}] // Union // Length;
Array[a, 100] (* Jean-François Alcover, Mar 24 2020 *)

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

A085312 Number of distinct 9th powers modulo n.

Original entry on oeis.org

1, 2, 3, 3, 5, 6, 3, 5, 3, 10, 11, 9, 5, 6, 15, 9, 17, 6, 3, 15, 9, 22, 23, 15, 21, 10, 3, 9, 29, 30, 11, 17, 33, 34, 15, 9, 5, 6, 15, 25, 41, 18, 15, 33, 15, 46, 47, 27, 15, 42, 51, 15, 53, 6, 55, 15, 9, 58, 59, 45, 21, 22, 9, 33, 25, 66, 23, 51, 69, 30, 71, 15, 9, 10, 63, 9, 33, 30, 27

Offset: 1

Labos Elemer, Jun 27 2003

Compare with enigmatic similarity of analogous odd-th power counts to A055653.

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

T. D. Noe, Table of n, a(n) for n = 1..1000
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

Cf. A000224[k=2], A046530[k=3], A052273[k=4], A052274[k=5], A052275[k=6], A085310[k=7], A085311[k=8], A085313[k=10], A085314[k=11], A228849[k=12], A055653.

A085312 := proc(m)
    {seq( modp(b^9,m),b=0..m-1) };
    nops(%) ;
end proc:
seq(A085312(m),m=1..100) ; # R. J. Mathar, Sep 22 2017

a[n_] := Table[PowerMod[i, 9, n], {i, 0, n - 1}] // Union // Length;
Array[a, 100] (* Jean-François Alcover, Mar 25 2020 *)

a(n)=my(f=factor(n)); prod(i=1, #f[, 1], my(k=f[i, 1]^f[i, 2]); #vecsort(vector(k, i, i^9%k), , 8)) \\ Charles R Greathouse IV, Sep 05 2013

A085313 Number of distinct 10th power residues modulo n.

Original entry on oeis.org

1, 2, 2, 2, 3, 4, 4, 2, 4, 6, 2, 4, 7, 8, 6, 3, 9, 8, 10, 6, 8, 4, 12, 4, 3, 14, 10, 8, 15, 12, 4, 5, 4, 18, 12, 8, 19, 20, 14, 6, 5, 16, 22, 4, 12, 24, 24, 6, 22, 6, 18, 14, 27, 20, 6, 8, 20, 30, 30, 12, 7, 8, 16, 9, 21, 8, 34, 18, 24, 24, 8, 8, 37, 38, 6, 20, 8, 28, 40, 9, 28, 10, 42, 16

Offset: 1

Labos Elemer, Jun 27 2003

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

T. D. Noe, Table of n, a(n) for n = 1..1000
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

Cf. A000224[k=2], A046530[k=3], A052273[k=4], A052274[k=5], A052275[k=6], A085310[k=7], A085311[k=8], A085312[k=9], A085314[k=11], A228849[k=12], A055653.

A085313 := proc(m)
    {seq( modp(b^10,m),b=0..m-1) };
    nops(%) ;
end proc:
seq(A085313(m),m=1..100) ; # R. J. Mathar, Sep 22 2017

a[n_] := Table[PowerMod[i, 10, n], {i, 0, n - 1}] // Union // Length;
Array[a, 100] (* Jean-François Alcover, Mar 25 2020 *)

a(n)=my(f=factor(n)); prod(i=1, #f[, 1], my(k=f[i, 1]^f[i, 2]); #vecsort(vector(k, i, i^10%k), , 8)) \\ Charles R Greathouse IV, Sep 05 2013

cp's OEIS Frontend

A123365 Values of k such that A046530(k) = (k+2)/3, where A046530(k) is the number of distinct residues of cubes mod k.

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A330695 Indices k at which record low values of the ratio A046530(k)/k occur.

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A000224 Number of squares mod n.

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A052273 Number of distinct 4th powers mod n.

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A052274 Number of distinct 5th powers mod n.

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A052275 Number of distinct 6th powers mod n.

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A085310 Number of distinct 7th powers modulo n.

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