A046530 -id:A046530 - cp's OEIS frontend

A085314 Number of distinct 11th powers modulo n.

1, 2, 3, 3, 5, 6, 7, 5, 7, 10, 11, 9, 13, 14, 15, 9, 17, 14, 19, 15, 21, 22, 3, 15, 21, 26, 19, 21, 29, 30, 31, 17, 33, 34, 35, 21, 37, 38, 39, 25, 41, 42, 43, 33, 35, 6, 47, 27, 43, 42, 51, 39, 53, 38, 55, 35, 57, 58, 59, 45, 61, 62, 49, 33, 65, 66, 7, 51, 9, 70, 71, 35, 73, 74, 63

Offset: 1

Labos Elemer, Jun 27 2003

Compare with enigmatic similarity of this and 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], A085312[k=9], A085313[k=10], A228849[k=12], A055653.

A085314 := proc(m)
    {seq( modp(b^11,m),b=0..m-1) };
    nops(%) ;
end proc:
seq(A085314(m),m=1..100) ; # R. J. Mathar, Sep 22 2017

a[n_] := Table[PowerMod[i, 11, 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^11%k), , 8)) \\ Charles R Greathouse IV, Sep 05 2013

A228849 Number of distinct 12th powers modulo n.

Original entry on oeis.org

1, 2, 2, 2, 2, 4, 2, 2, 2, 4, 6, 4, 2, 4, 4, 2, 5, 4, 4, 4, 4, 12, 12, 4, 6, 4, 4, 4, 8, 8, 6, 3, 12, 10, 4, 4, 4, 8, 4, 4, 11, 8, 8, 12, 4, 24, 24, 4, 8, 12, 10, 4, 14, 8, 12, 4, 8, 16, 30, 8, 6, 12, 4, 5, 4, 24, 12, 10, 24, 8, 36, 4, 7, 8, 12, 8, 12, 8, 14, 4, 10

Offset: 1

Arkadiusz Wesolowski, Sep 05 2013

R. J. Mathar, Table of n, a(n) for n = 1..10000

Cf. A000224 (squares), 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).

[#Set([k^12 mod n : k in [1..n]]) : n in [1..81]];

A228849 := proc(n)
    {seq(i^12 mod n, i=0..n-1)} ;
    nops(%) ;
end proc: # R. J. Mathar, Sep 21 2017

a[n_] := Table[PowerMod[i, 12, 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^12%k), , 8)) \\ Charles R Greathouse IV, Sep 05 2013

A046630 Number of cubic residues mod 2^n.

Original entry on oeis.org

1, 2, 3, 5, 10, 19, 37, 74, 147, 293, 586, 1171, 2341, 4682, 9363, 18725, 37450, 74899, 149797, 299594, 599187, 1198373, 2396746, 4793491, 9586981, 19173962, 38347923, 76695845, 153391690, 306783379, 613566757, 1227133514, 2454267027

Offset: 0

David W. Wilson

			For n=3, the cubes 0^3, 1^3, 2^3, ..., 7^3 reduced mod 2^3 = 8 are 0,1,0,3,0,5,0,7, five different values, so a(3)=5. - _N. J. A. Sloane_, Sep 30 2018

S. R. Finch and Pascal Sebah, Squares and Cubes Modulo n, arXiv:math/0604465 [math.NT], 2006-2016.
Index entries for linear recurrences with constant coefficients, signature (2,0,1,-2).

Cf. A033138.

A049347 := proc(n) op( (n mod 3)+1,[1,-1,0]) ;end proc:
A046630 := proc(n) 2^(n+2)/7+2/3-(5*A049347(n)+A049347(n-1))/21 ; end proc: # R. J. Mathar, Feb 27 2011

LinearRecurrence[{2, 0, 1, -2}, {1, 2, 3, 5}, 33] (* Jean-François Alcover, Nov 17 2017 *)

a(n)=(4<Charles R Greathouse IV, Jan 03 2013

a(n) = ceiling(2^(n+2)/7) [Finch-Sebah, page 12]. - N. J. A. Sloane, Sep 30 2018
G.f.: (-2*x^3-x^2+1)/((1-2*x)*(1-x^3)).
a(n) = A046530(2^n) = 2^(n+2)/7 + 2/3 - (5*A049347(n)+A049347(n-1))/21. - R. J. Mathar, Feb 27 2011
a(n) = 1 + A033138(n) for n >= 1. - John Keith, Mar 07 2022

A046631 Number of cubic residues mod 3^n.

Original entry on oeis.org

1, 3, 3, 7, 21, 57, 169, 507, 1515, 4543, 13629, 40881, 122641, 367923, 1103763, 3311287, 9933861, 29801577, 89404729, 268214187, 804642555, 2413927663, 7241782989, 21725348961, 65176046881, 195528140643, 586584421923

Offset: 0

David W. Wilson

S. R. Finch and Pascal Sebah, Squares and Cubes Modulo n, arXiv:math/0604465 [math.NT], 2006-2016.
Index entries for linear recurrences with constant coefficients, signature (3,0,1,-3).

A049347 := proc(n) op( (n mod 3)+1,[1,-1,0]) ; end proc:
A046631 := proc(n) 3^(n+1)/13+4/3-(22*A049347(n)-16*A049347(n-1))/39 ; end proc: # R. J. Mathar, Feb 27 2011

LinearRecurrence[{3, 0, 1, -3}, {1, 3, 3, 7}, 27] (* Jean-François Alcover, Nov 22 2017 *)

G.f.: (-3*x^3-6*x^2+1)/((1-3*x)*(1-x^3)).

a(n) = A046530(3^n) = 4/3 + 3^(n+1)/13 - (22*A049347(n) - 16*A049347(n-1))/39. - R. J. Mathar, Feb 27 2011

A087786 a(n) = number of solutions to x^3 - y^3 == 0 (mod n).

Original entry on oeis.org

1, 2, 3, 6, 5, 6, 19, 20, 27, 10, 11, 18, 37, 38, 15, 40, 17, 54, 55, 30, 57, 22, 23, 60, 45, 74, 135, 114, 29, 30, 91, 112, 33, 34, 95, 162, 109, 110, 111, 100, 41, 114, 127, 66, 135, 46, 47, 120, 175, 90, 51, 222, 53, 270, 55, 380, 165, 58, 59, 90, 181, 182, 513, 352, 185

Offset: 1

Yuval Dekel (dekelyuval(AT)hotmail.com), Oct 06 2003

Andrew Howroyd, Table of n, a(n) for n = 1..10000

Cf. A046530, A087694, A062803.

a(n)={my(v=vector(n)); for(i=0, n-1, v[i^3%n + 1]++); sum(i=0, n-1, v[i+1]^2)} \\ Andrew Howroyd, Jul 17 2018

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

Multiplicative with a(p^e) = p^(2*floor(2*e/3)) + Sum_{i=0..floor((e-1)/3)} k*(p-1)*p^(e+i-1) where k = 3 if (p = 3 and 3*i+1 = e) or (p mod 3 = 1) otherwise k = 1. - Andrew Howroyd, Jul 17 2018

More terms from John W. Layman, Oct 18 2003

A257301 Number of cubic nonresidues modulo n.

Original entry on oeis.org

0, 0, 0, 1, 0, 0, 4, 3, 6, 0, 0, 3, 8, 8, 0, 6, 0, 12, 12, 5, 12, 0, 0, 9, 4, 16, 20, 19, 0, 0, 20, 13, 0, 0, 20, 27, 24, 24, 24, 15, 0, 24, 28, 11, 30, 0, 0, 18, 34, 8, 0, 37, 0, 40, 0, 41, 36, 0, 0, 15, 40, 40, 54, 27, 40, 0, 44, 17, 0, 40, 0, 57, 48, 48, 12, 55, 44, 48, 52, 30

Offset: 1

Stanislav Sykora, Apr 19 2015

nonn

a(n) is the number of values r, 0<=r=0, (m^p)%n != r. Compared to quadratic nonresidues (p=2, sequence A095972), the most evident difference is the frequent occurrence of a(n)=0 (for values of n which belong to A074243).

			a(5)=0, because the set {(k^3)%5}, with k=0..4, evaluates to {0,1,3,2,4},
        with no missing residue values.
a(7)=4, because the set {(k^3)%7}, with k=0..6, evaluates to
        {0,1,1,6,1,6,6}, with missing residue values {2,3,4,5}.

Stanislav Sykora, Table of n, a(n) for n = 1..10000

Nonresidues for other exponents: A095972 (p=2), A257302 (p=4), A257303 (p=5).

Cf. A074243, A046530.

seq(n - nops({seq(a^3 mod n,a=0..n-1)}), n=1..100); # Robert Israel, Apr 20 2015

Table[Length[Complement[Range[n - 1], Union[Mod[Range[n]^3, n]]]], {n, 100}] (* Vincenzo Librandi, Apr 20 2015 *)

nrespowp(n,p) = {my(v=vector(n),d=0);
  for(r=0,n-1,v[1+(r^p)%n]+=1);
  for(k=1,n,if(v[k]==0,d++));
  return(d);}
a(n) = nrespowp(n,3)

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)); n-prod(i=1, #f~, g(f[i,1], f[i,2])) \\ Charles R Greathouse IV, Apr 20 2015

a(n) = n - A046530(n).
Satisfies a(A074243(n))=0.
Satisfies a(n) <= n-3 (residues 0, 1, and n-1 are always present).
a(n) = n - A046530(n). - Robert Israel, Apr 20 2015

A376202 Number of pairs 1 <= x <= y <= n-1 such that gcd(x,n) = gcd(y,n) = gcd(x+y,n) = 1 and 1/x + 1/y == 1/(x+y) mod n.

Original entry on oeis.org

0, 0, 2, 0, 0, 0, 6, 0, 0, 0, 0, 0, 12, 0, 0, 0, 0, 0, 18, 0, 12, 0, 0, 0, 0, 0, 0, 0, 0, 0, 30, 0, 0, 0, 0, 0, 36, 0, 24, 0, 0, 0, 42, 0, 0, 0, 0, 0, 42, 0, 0, 0, 0, 0, 0, 0, 36, 0, 0, 0, 60, 0, 0, 0, 0, 0, 66, 0, 0, 0, 0, 0, 72, 0, 0, 0, 0, 0, 78, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 144, 0, 60, 0, 0, 0, 96, 0, 0, 0, 0, 0, 102, 0, 0

Offset: 1

Tom Duff and N. J. A. Sloane, Oct 06 2024

nonn

In general, 1/x + 1/y = 1/(x+y) is the wrong way to add fractions!
See A376203 for a(2*n-1)/2 and A376755 for a(6*n+1)/6.
From Robert Israel, Nov 06 2024: (Start)
If a(n) = 0 then a(m) = 0 whenever m is a multiple of n.
It appears that the primes p for which a(p) > 0 are A007645. (End)

			For n = 3 the a(3) = 2 solutions are (x,y) = (1,1) and (2,2).
For n = 7 the a(7) = 6 solutions are (x,y) = (1,2), (1,4), (2,4), (3,5), (3,6), (5,6).

Robert Israel, Table of n, a(n) for n = 1..6000

Cf. A000086, A007645, A046530, A290731, A376203, A376755, A376756, A376757.

a:=[];
for n from 1 to 140 do
c:=0;
for y from 1 to n-1 do
for x from 1 to y do
if gcd(y,n) = 1 and gcd(x,n) = 1 and gcd(x+y,n) = 1  and (1/x + 1/y - 1/(x+y)) mod n = 0 then c:=c+1; fi;
od: # od x
od: # od y
a:=[op(a),c];
od: # od n
a;

from math import gcd
def A376202(n):
    c = 0
    for x in range(1,n):
        if gcd(x,n) == 1:
            for y in range(x,n):
                if gcd(y,n)==gcd(z:=x+y,n)==1 and not (w:=z**2-x*y)//gcd(w,x*y*z)%n:
                    c += 1
    return c # Chai Wah Wu, Oct 06 2024

A376203 a(n) = A376202(2*n-1)/2.

Original entry on oeis.org

0, 1, 0, 3, 0, 0, 6, 0, 0, 9, 6, 0, 0, 0, 0, 15, 0, 0, 18, 12, 0, 21, 0, 0, 21, 0, 0, 0, 18, 0, 30, 0, 0, 33, 0, 0, 36, 0, 0, 39, 0, 0, 0, 0, 0, 72, 30, 0, 48, 0, 0, 51, 0, 0, 54, 36, 0, 0, 0, 0, 0, 0, 0, 63, 42, 0, 108, 0, 0, 69

Offset: 1

Tom Duff and N. J. A. Sloane, Oct 06 2024

nonn

Chai Wah Wu, Table of n, a(n) for n = 1..7814

Cf. A000086, A087786, A046530, A290731, A376202, A376755, A376756, A376757.

from math import gcd
def A376203(n):
    c, m = 0, (n<<1)-1
    for x in range(1,m):
        if gcd(x,m) == 1:
            for y in range(x,m):
                if gcd(y,m)==gcd(z:=x+y,m)==1 and not (w:=z**2-x*y)//gcd(w,x*y*z)%m:
                    c += 1
    return c>>1 # Chai Wah Wu, Oct 06 2024

A376755 a(n) = A376202(6*n+1)/6.

Original entry on oeis.org

1, 2, 3, 0, 5, 6, 7, 7, 0, 10, 11, 12, 13, 0, 24, 16, 17, 18, 0, 0, 21, 36, 23, 0, 25, 26, 27, 26, 0, 30, 0, 32, 33, 0, 35, 60, 37, 38, 0, 40, 72, 0, 72, 0, 45, 46, 47, 0, 0, 84, 51, 52, 0, 0, 55, 56, 49, 58, 0, 57, 61, 62, 63, 0, 0, 66, 120, 68, 0, 70, 120, 72

Offset: 1

Tom Duff and N. J. A. Sloane, Oct 06 2024

nonn

Chai Wah Wu, Table of n, a(n) for n = 1..3470

Cf. A000086, A046530, A087786, A290731, A376202, A376203, A376756, A376757.

from math import gcd
def A376755(n):
    c, m = 0, 6*n|1
    for x in range(1,m):
        if gcd(x,m) == 1:
            for y in range(x,m):
                if gcd(y,m)==gcd(z:=x+y,m)==1 and not (w:=z**2-x*y)//gcd(w,x*y*z)%m:
                    c += 1
    return c//6 # Chai Wah Wu, Oct 06 2024

a(51)-a(72) from Chai Wah Wu, Oct 06 2024

A376756 Number of pairs 0 <= x <= y <= n-1 such that x^2 + x*y + y^2 == 0 (mod n).

Original entry on oeis.org

1, 1, 3, 3, 1, 3, 7, 3, 6, 1, 1, 9, 13, 7, 3, 10, 1, 6, 19, 3, 21, 1, 1, 9, 15, 13, 18, 27, 1, 3, 31, 10, 3, 1, 7, 21, 37, 19, 39, 3, 1, 21, 43, 3, 6, 1, 1, 30, 70, 15, 3, 51, 1, 18, 1, 27, 57, 1, 1, 9, 61, 31, 60, 36, 13, 3, 67, 3, 3, 7, 1, 21, 73, 37, 45, 75, 7, 39, 79, 10, 45, 1, 1, 81, 1, 43, 3, 3, 1, 6, 163, 3, 93, 1, 19, 30, 97

Offset: 1

Tom Duff and N. J. A. Sloane, Oct 06 2024

nonn

Seiichi Manyama, Table of n, a(n) for n = 1..10000

Cf. A000086, A046530, A087786, A290731, A376202, A376203, A376755, A376757.

def A376756(n):
    c = 0
    for x in range(n):
        z = x**2%n
        for y in range(x,n):
            if not (z+y*(x+y))%n:
                c += 1
    return c # Chai Wah Wu, Oct 06 2024

cp's OEIS Frontend

A085314 Number of distinct 11th powers modulo n.

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A228849 Number of distinct 12th powers modulo n.

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A046630 Number of cubic residues mod 2^n.

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A046631 Number of cubic residues mod 3^n.

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A087786 a(n) = number of solutions to x^3 - y^3 == 0 (mod n).

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A257301 Number of cubic nonresidues modulo n.

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A376202 Number of pairs 1 <= x <= y <= n-1 such that gcd(x,n) = gcd(y,n) = gcd(x+y,n) = 1 and 1/x + 1/y == 1/(x+y) mod n.

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