A250207 -id:A250207 - cp's OEIS frontend

A293482 The number of 5th powers in the multiplicative group modulo n.

1, 1, 2, 2, 4, 2, 6, 4, 6, 4, 2, 4, 12, 6, 8, 8, 16, 6, 18, 8, 12, 2, 22, 8, 4, 12, 18, 12, 28, 8, 6, 16, 4, 16, 24, 12, 36, 18, 24, 16, 8, 12, 42, 4, 24, 22, 46, 16, 42, 4, 32, 24, 52, 18, 8, 24, 36, 28, 58, 16, 12, 6, 36, 32, 48, 4, 66, 32, 44, 24, 14, 24, 72, 36, 8, 36, 12, 24, 78, 32, 54, 8, 82, 24

Offset: 1

R. J. Mathar, Oct 10 2017

The size of the set of numbers j^5 mod n, gcd(j,n)=1, 1 <= j <= n.

A000010(n) / a(n) is another multiplicative integer sequence.

R. J. Mathar, Table of n, a(n) for n = 1..7548
R. J. Mathar, Size of the set of residues of integer powers of fixed exponent, 2017.
Samer Seraj, Counting general power residues, Not. Numb. Th. Discr. Math. 28 (4) (2022) 730-743.
Samer Seraj, Resolution of Mathar's conjectures on counting power residues, INTEGERS 23 (2023) #A62.

The number of k-th powers in the multiplicative group modulo n: A046073 (k=2), A087692 (k=3), A250207 (k=4), this sequence (k=5), A293483 (k=6), A293484 (k=7), A293485 (k=8).

Cf. A052274, A319099, A000010.

A293482 := proc(n)
    local r,j;
    r := {} ;
    for j from 1 to n do
        if igcd(j,n)= 1 then
            r := r union { modp(j &^ 5,n) } ;
        end if;
    end do:
    nops(r) ;
end proc:
seq(A293482(n),n=1..120) ;

a[n_] := Module[{r, j}, r = {}; For[j = 1, j <= n, j++, If[GCD[j, n] == 1, r = r ~Union~ {PowerMod[j, 5, n]}] ]; Length[r]];
Table[a[n], {n, 1, 120}] (* Jean-François Alcover, Feb 14 2023, after R. J. Mathar *)
f[p_, e_] := (p - 1)*p^(e - 1)/If[Mod[p, 5] == 1, 5, 1]; f[2, e_] := 2^(e - 1); f[2, 1] = 1; f[5, e_] := 4*5^(e-2); f[5, 1] = 4; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Aug 10 2023 *)

Conjecture: a(2^e) = 1 for e <= 1; a(2^e) = 2^(e-1) for e >= 1; a(5)=4; a(5^e) = 4*5^(e-2) for e > 1; a(p^e) = (p-1)*p^(e-1) for p == {2,3,4} (mod 5); a(p^e) = (p-1)*p^(e-1)/5 for p == 1 (mod 5). - R. J. Mathar, Oct 13 2017

a(n) = A000010(n)/A319099(n). This implies that the conjecture above is true. - Jianing Song, Nov 10 2019

A087692 Number of cubes in multiplicative group modulo n.

Original entry on oeis.org

1, 1, 2, 2, 4, 2, 2, 4, 2, 4, 10, 4, 4, 2, 8, 8, 16, 2, 6, 8, 4, 10, 22, 8, 20, 4, 6, 4, 28, 8, 10, 16, 20, 16, 8, 4, 12, 6, 8, 16, 40, 4, 14, 20, 8, 22, 46, 16, 14, 20, 32, 8, 52, 6, 40, 8, 12, 28, 58, 16, 20, 10, 4, 32, 16, 20, 22, 32, 44, 8, 70, 8, 24, 12, 40, 12, 20, 8, 26, 32, 18, 40

Offset: 1

Yuval Dekel (dekelyuval(AT)hotmail.com), Sep 27 2003

Cubic analog of A046073. - Steven Finch, Mar 01 2006

Robert Israel, Table of n, a(n) for n = 1..10000
Steven R. Finch and Pascal Sebah, Squares and Cubes Modulo n, arXiv:math/0604465 [math.NT], 2006-2016.

Cf. A000010, A060839, A046073 (squares), A250207 (4th powers).

b:= proc(p,i)
  if p = 3 then if i=1 then 2 else 2*3^(i-2) fi
  elif p mod 6 = 1 then (p-1)*p^(i-1)/3
  else (p-1)*p^(i-1)
  fi
end proc:
seq(mul(b(f[1],f[2]), f = ifactors(n)[2]), n = 1 .. 1000); # Robert Israel, Jan 04 2015

Map[Length,Table[Select[Range[n],CoprimeQ[#, n] && IntegerQ[PowerMod[#, 1/3, n]] &], {n, 1, 82}]] (* Geoffrey Critzer, Jan 07 2015 *)
f[p_, e_] := (p - 1)*p^(e - 1)/If[Mod[p, 6] == 1, 3, 1]; f[3, e_] := 2*3^(e-2); f[3, 1] = 2; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Aug 10 2023 *)

a(n) = my(f = factor(n)); prod(j=1, #f~, p=f[j,1]; k=f[j,2]; if (p == 3, if (k==1, 2, 2*3^(k-2)), if ((p % 6) == 1, ((p-1)*p^(k-1))/3, (p-1)*p^(k-1)))); \\ Michel Marcus, Jan 05 2015

a(n) = phi(n) / A060839(n).
Multiplicative with a(3) = 2, a(3^k) = 2*3^(k-2) otherwise;
a(p^k) = (p-1)*p^(k-1)/3 if prime p == 1 mod 6; a(p^k) = (p-1)*p^(k-1) for all other primes p. - Robert Israel, Jan 04 2015
Sum_{k=1..n} a(k) ~ c * n^2/log(n)^(1/3), where c = (17/(36*Gamma(2/3))) * Product_{p = 3 or p prime == 2 (mod 3)} (1+1/*p)*(1-1/p)^(2/3) * Product_{p prime == 1 (mod 3)} (1+1/(3*p))*(1-1/p)^(2/3) = 0.33051128776333262024... (Finch and Sebah, 2006). - Amiram Eldar, Oct 18 2022

More terms from Steven Finch, Mar 01 2006

A293483 The number of 6th powers in the multiplicative group modulo n.

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 1, 1, 1, 2, 5, 1, 2, 1, 2, 2, 8, 1, 3, 2, 1, 5, 11, 1, 10, 2, 3, 1, 14, 2, 5, 4, 5, 8, 2, 1, 6, 3, 2, 2, 20, 1, 7, 5, 2, 11, 23, 2, 7, 10, 8, 2, 26, 3, 10, 1, 3, 14, 29, 2, 10, 5, 1, 8, 4, 5, 11, 8, 11, 2, 35, 1, 12, 6, 10, 3, 5, 2, 13, 4, 9, 20

Offset: 1

R. J. Mathar, Oct 10 2017

The size of the set of numbers j^6 mod n, gcd(j,n)=1, 1 <= j <= n.

A000010(n) / a(n) is another multiplicative integer sequence.

R. J. Mathar, Table of n, a(n) for n = 1..10132
Richard J. Mathar, Size of the Set of Residues of Integer Powers of Fixed Exponent, 2017.

The number of k-th powers in the multiplicative group modulo n: A046073 (k=2), A087692 (k=3), A250207 (k=4), A293482 (k=5), this sequence (k=6), A293484 (k=7), A293485 (k=8).

Cf. A052275, A319100, A000010.

A293483 := proc(n)
    local r,j;
    r := {} ;
    for j from 1 to n do
        if igcd(j,n)= 1 then
            r := r union { modp(j &^ 6,n) } ;
        end if;
    end do:
    nops(r) ;
end proc:
seq(A293483(n),n=1..120) ;

a[n_] := EulerPhi[n]/Count[Range[0, n - 1]^6 - 1, k_ /; Divisible[k, n]];
Array[a, 100] (* Jean-François Alcover, May 24 2023 *)
f[p_, e_] := (p - 1)*p^(e - 1)/If[Mod[p, 6] == 1, 6, 2]; f[2, e_] := If[e <= 3, 1, 2^(e - 3)]; f[3, e_] := If[e <= 2, 1, 3^(e - 2)]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Aug 10 2023 *)

Conjecture: a(2^e) = 1 for e <= 3; a(2^e) = 2^(e-3) for e >= 3; a(3^e) = 1 for e <= 2; a(3^e) = 3^(e-2) for e >= 2; a(p^e) = (p-1)*p^(e-1)/2 for p == 5 (mod 6); a(p^e) = (p-1)*p^(e-1)/6 for p == 1 (mod 6). - R. J. Mathar, Oct 13 2017

a(n) = A000010(n)/A319100(n). This implies that the conjecture above is true. - Jianing Song, Nov 10 2019

A293484 The number of 7th powers in the multiplicative group modulo n.

Original entry on oeis.org

1, 1, 2, 2, 4, 2, 6, 4, 6, 4, 10, 4, 12, 6, 8, 8, 16, 6, 18, 8, 12, 10, 22, 8, 20, 12, 18, 12, 4, 8, 30, 16, 20, 16, 24, 12, 36, 18, 24, 16, 40, 12, 6, 20, 24, 22, 46, 16, 6, 20, 32, 24, 52, 18, 40, 24, 36, 4, 58, 16, 60, 30, 36, 32, 48, 20, 66, 32, 44, 24, 10, 24, 72, 36, 40, 36

Offset: 1

R. J. Mathar, Oct 10 2017

The size of the set of numbers j^7 mod n, gcd(j,n)=1, 1 <= j <= n.

A000010(n) / a(n) is another multiplicative integer sequence (size of the kernel of the isomorphism of the multiplicative group modulo n to the multiplicative group of 7th powers modulo n).

R. J. Mathar, Table of n, a(n) for n = 1..10116
Richard J. Mathar, Size of the Set of Residues of Integer Powers of Fixed Exponent, 2017.

The number of k-th powers in the multiplicative group modulo n: A046073 (k=2), A087692 (k=3), A250207 (k=4), A293482 (k=5), A293483 (k=6), this sequence (k=7), A293485 (k=8).

Cf. A085310, A319101, A000010.

A293484 := proc(n)
    local r,j;
    r := {} ;
    for j from 1 to n do
        if igcd(j,n)= 1 then
            r := r union { modp(j &^ 7,n) } ;
        end if;
    end do:
    nops(r) ;
end proc:
seq(A293484(n),n=1..120) ;

a[n_] := EulerPhi[n]/Count[Range[0, n - 1]^7 - 1, k_ /; Divisible[k, n]];
Array[a, 100] (* Jean-François Alcover, May 24 2023 *)
f[p_, e_] := (p-1)*p^(e-1)/If[Mod[p, 7] == 1, 7, 1]; f[2, e_] := 2^(e-1); f[7, 1] = 6; f[7, e_] := 6*7^(e-2); a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Aug 10 2023 *)

Conjecture: a(2^e) = 1 for e <= 1; a(2^e) = 2^(e-1) for e >= 1; a(7^e) = 6 for e=1; a(7^e) = 6*7^(e-2) for e >= 2; a(p^e) = (p-1)*p^(e-1) for p == {2,3,4,5,6} (mod 7); a(p^e) = (p-1)*p^(e-1)/7 for p == 1 (mod 7). - R. J. Mathar, Oct 13 2017

a(n) = A000010(n)/A319101(n). This implies that the conjecture above is true. - Jianing Song, Nov 10 2019

A293485 The number of 8th powers in the multiplicative group modulo n.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 3, 1, 3, 1, 5, 1, 3, 3, 1, 1, 2, 3, 9, 1, 3, 5, 11, 1, 5, 3, 9, 3, 7, 1, 15, 1, 5, 2, 3, 3, 9, 9, 3, 1, 5, 3, 21, 5, 3, 11, 23, 1, 21, 5, 2, 3, 13, 9, 5, 3, 9, 7, 29, 1, 15, 15, 9, 2, 3, 5, 33, 2, 11, 3, 35, 3, 9, 9, 5, 9, 15, 3, 39, 1, 27, 5, 41, 3, 2

Offset: 1

R. J. Mathar, Oct 10 2017

The size of the set of numbers j^8 mod n, gcd(j,n)=1, 1<=j<=n.

Antti Karttunen, Table of n, a(n) for n = 1..16384
Richard J. Mathar, Size of the Set of Residues of Integer Powers of Fixed Exponent, research paper, 2017.

The number of k-th powers in the multiplicative group modulo n: A046073 (k=2), A087692 (k=3), A250207 (k=4), A293482 (k=5), A293483 (k=6), A293484 (k=7), this sequence (k=8).

Cf. A085311, A247257 (order of the kernel isomorphism of Z/nZ to this group), A000010.

A293485 := proc(n)
    local r,j;
    r := {} ;
    for j from 1 to n do
        if igcd(j,n)= 1 then
            r := r union { modp(j &^ 8,n) } ;
        end if;
    end do:
    nops(r) ;
end proc:
seq(A293485(n),n=1..120) ;

a[n_] := EulerPhi[n]/Count[Range[0, n - 1]^8 - 1, k_ /; Divisible[k, n]];
Array[a, 100] (* Jean-François Alcover, May 24 2023 *)
f[p_, e_] := (p - 1)*p^(e - 1)/Switch[Mod[p, 8], 1, 8, 5, 4, , 2]; f[2, e] := If[e <= 4, 1, 2^(e - 5)]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Aug 10 2023 *)

\\ The following two functions by Charles R Greathouse IV, from A247257:
g(p, e) = if(p==2, 2^min(e-1, 4), if(p%4==3, 2, if(p%8==5, 4, 8)));
A247257(n) = my(f=factor(n)); prod(i=1, #f~, g(f[i, 1], f[i, 2]));
A293485(n) = (eulerphi(n)/A247257(n)); \\ Antti Karttunen, Dec 05 2017

A000010(n) / a(n) = A247257(n).

Multiplicative with a(2^e) = 1 for e<=4, a(2^e) = 2^(e-5) for e>=5; a(p^e) = (p-1)*p^(e-1)/8 for p == 1 (mod 8); a(p^e) = (p-1)*p^(e-1)/4 for p == 5 (mod 8); a(p^e) = (p-1)*p^(e-1)/2 for p == {3,7} (mod 8). - R. J. Mathar, Oct 15 2017 [corrected by Georg Fischer, Jul 21 2022]

cp's OEIS Frontend

A293482 The number of 5th powers in the multiplicative group modulo n.

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A087692 Number of cubes in multiplicative group modulo n.

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A293483 The number of 6th powers in the multiplicative group modulo n.

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A293484 The number of 7th powers in the multiplicative group modulo n.

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A293485 The number of 8th powers in the multiplicative group modulo n.

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