A239441 -id:A239441 - cp's OEIS frontend

A079458 Number of Gaussian integers in a reduced system modulo n.

1, 2, 8, 8, 16, 16, 48, 32, 72, 32, 120, 64, 144, 96, 128, 128, 256, 144, 360, 128, 384, 240, 528, 256, 400, 288, 648, 384, 784, 256, 960, 512, 960, 512, 768, 576, 1296, 720, 1152, 512, 1600, 768, 1848, 960, 1152, 1056, 2208, 1024, 2352, 800, 2048, 1152, 2704

Offset: 1

Vladeta Jovovic, Jan 14 2003

Number of units in the ring consisting of the Gaussian integers modulo n. - Jason Kimberley, Dec 07 2015

			{1, i, 1+2i, 2+i, 3, 3i, 3+2i, 2+3i} is the set of eight units in the Gaussian integers modulo 4. - _Jason Kimberley_, Dec 07 2015

Jason Kimberley, Table of n, a(n) for n = 1..10000
Jonathan M. Borwein, Adventures with the OEIS: Five sequences Tony may like, Guttmann 70th [Birthday] Meeting, 2015, revised May 2016. [Wayback Machine link]
Jonathan M. Borwein, Adventures with the OEIS: Five sequences Tony may like, Guttmann 70th [Birthday] Meeting, 2015, revised May 2016. [Cached copy, with permission]
Catalina Calderón, Jose Maria Grau, A. Oller-Marcén, and László Tóth, Counting invertible sums of squares modulo n and a new generalization of Euler's totient function, Publicationes Mathematicae-Debrecen, Vol. 87 (1-2) (2015), pp. 133-145; arXiv preprint, arXiv:1403.7878 [math.NT], 2014.

Equals four times A218147. - Jason Kimberley, Nov 14 2015
Sequences giving the number of solutions to the equation GCD(x_1^2+...+x_k^2, n) = 1 with 0 < x_i <= n: A000010 (k=1), A079458 (k=2), A053191 (k=3), A227499 (k=4), A238533 (k=5), A238534 (k=6), A239442 (k=7), A239441 (k=8), A239443 (k=9).
Cf. A003557, A007947, A204617, A062570.
Equivalent of arithmetic functions in the ring of Gaussian integers (the corresponding functions in the ring of integers are in the parentheses): A062327 ("d", A000005), A317797 ("sigma", A000203), this sequence ("phi", A000010), A227334 ("psi", A002322), A086275 ("omega", A001221), A078458 ("Omega", A001222), A318608 ("mu", A008683).
Equivalent in the ring of Eisenstein integers: A319445.

A079458 := func)>; // Jason Kimberley, Nov 14 2015

with(GaussInt): seq(GIphi(n), n=1..100);

phi[1]=1;phi[p_, s_] := Which[Mod[p, 4] == 3, p^(2 s - 2) (p^2 - 1), Mod[p, 4] == 1, p^(2 s - 2) ((p - 1))^2, True, 2^(2 s - 1)];phi[n_] := Product[phi[FactorInteger[n][[i, 1]], FactorInteger[n][[i, 2]]], {i, Length[FactorInteger[n]]}];Table[phi[n], {n, 1, 33}] (* José María Grau Ribas, Mar 16 2014 *)
f[p_, e_] := (p - 1)*p^(2*e - 1) * If[p == 2, 1, 1 - (-1)^((p-1)/2)/p]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Feb 13 2024 *)

a(n)=
{
    my(r=1, f=factor(n));
    for(j=1, #f[, 1], my(p=f[j, 1], e=f[j, 2]);
        if(p==2, r*=2^(2*e-1));
        if(p%4==1, r*=(p-1)^2*p^(2*e-2));
        if(p%4==3, r*=(p^2-1)*p^(2*e-2));
    );
    return(r);
} \\ Jianing Song, Sep 16 2018

Multiplicative with a(2^e) = 2^(2*e-1), a(p^e) = (p^2-1)*p^(2*e-2) if p mod 4=3 and a(p^e) = (p-1)^2*p^(2*e-2) if p mod 4=1.
a(n) = A003557(n)^2 * a(A007947(n)), where a(2)=2, a(p)=(p-1)^2 for prime p=1(mod 4), a(p)=p^2-1 for prime p=3(mod 4), and a(n*m)=a(n)*a(m) for n coprime to m. - Jason Kimberley, Nov 16 2015
From Amiram Eldar, Feb 13 2024: (Start)
Dirichlet g.f.: zeta(s-2) * (1 - 1/2^(s-1)) * Product_{p prime > 2} (1 - 1/p^(s-1) - (-1)^((p-1)/2)*(p-1)/p^s).
Sum_{k=1..n} a(k) = c * n^3 / 3 + O(n^2 * log(n)), where c = (3/4) * Product_{p prime > 2} (1 - 1/p^2 - (-1)^((p-1)/2)*(p-1)/p^3) = (3/4) * A334427 * Product_{p prime == 1 (mod 4)} (1 - 2/p^2 + 1/p^3) = 0.6498027559... (Calderón et al., 2015). (End)
a(n) = A204617(n)*A062570(n). - Ridouane Oudra, Jun 05 2024

A239442 a(n) = phi(n^7).

Original entry on oeis.org

1, 64, 1458, 8192, 62500, 93312, 705894, 1048576, 3188646, 4000000, 17715610, 11943936, 57921708, 45177216, 91125000, 134217728, 386201104, 204073344, 846825858, 512000000, 1029193452, 1133799040, 3256789558, 1528823808, 4882812500, 3706989312, 6973568802, 5782683648, 16655052988

Offset: 1

José María Grau Ribas, Mar 19 2014

Number of solutions of the equation gcd(x_1^2 + ... + x_7^2, n)=1 with 0 < x_i <= n.

Seiichi Manyama, Table of n, a(n) for n = 1..10000
C. Calderón, J. M. Grau, A. Oller-Marcén, and László Tóth, Counting invertible sums of squares modulo n and a new generalization of Euler totient function, arXiv:1403.7878 [math.NT], 2014.

Defining Phi_k(n):= number of solutions of the equation gcd(x_1^2 + ... + x_k^2, n) = 1 with 0 < x_i <= n.
Phi_1(n) = phi(n) = A000010.
Phi_2(n) = A079458.
Phi_3(n) = phi(n^3) = n^2*phi(n)= A053191.
Phi_4(n) = A227499.
Phi_5(n) = phi(n^5) = n^4*phi(n)= A238533.
Phi_6(n) = A238534.
Phi_7(n) = phi(n^7) = n^6*phi(n)= A239442.
Phi_8(n) = A239441.
Phi_9(n) = phi(n^9) = n^8*phi(n)= A239443.

with(numtheory); A239442:=n->phi(n^7); seq(A239442(n), n=1..100); # Wesley Ivan Hurt, Apr 01 2014

Mathematica
```
Table[EulerPhi[n^7], {n, 100}]
```

a(n) = n^6*eulerphi(n); \\ Michel Marcus, Mar 10 2018

a(n) = n^6*phi(n).
Dirichlet g.f.: zeta(s - 7) / zeta(s - 6). The n-th term of the Dirichlet inverse is n^6 * A023900(n) = (-1)^omega(n) * a(n) / A003557(n), where omega=A001221. - Álvar Ibeas, Nov 24 2017
Sum_{k=1..n} a(k) ~ 3*n^8 / (4*Pi^2). - Vaclav Kotesovec, Feb 02 2019
Sum_{n>=1} 1/a(n) = Product_{p prime} (1 + p/(p^8 - p^7 - p + 1)) = 1.01646280485545934937... - Amiram Eldar, Dec 06 2020

A239443 a(n) = phi(n^9), where phi = A000010.

Original entry on oeis.org

1, 256, 13122, 131072, 1562500, 3359232, 34588806, 67108864, 258280326, 400000000, 2143588810, 1719926784, 9788768652, 8854734336, 20503125000, 34359738368, 111612119056, 66119763456, 305704134738, 204800000000, 453874312332, 548758735360, 1722841676182, 880602513408, 3051757812500

Offset: 1

José María Grau Ribas, Mar 22 2014

Number of solutions of the equation GCD(x_1^2 + ... + x_9^2,n)=1 with 0 < x_i <= n.

In general, for m>0, Sum_{k=1..n} phi(k^m) ~ 6 * n^(m+1) / ((m+1)*Pi^2). - Vaclav Kotesovec, Feb 02 2019

Seiichi Manyama, Table of n, a(n) for n = 1..10000
C. Calderón, J. M. Grau, A. Oller-Marcén, and László Tóth, Counting invertible sums of squares modulo n and a new generalization of Euler totient function, arXiv:1403.7878 [math.NT], 2014.

Defining Phi_k(n):= number of solutions of the equation GCD(x_1^2 + ... + x_k^2,n)=1 with 0 < x_i <= n.
Phi_1(n) = phi(n) = A000010(n).
Phi_2(n) = A079458(n).
Phi_3(n) = phi(n^3) = n^2*phi(n)= A053191(n).
Phi_4(n) = A227499(n).
Phi_5(n) = phi(n^5) = n^4*phi(n)= A238533(n).
Phi_6(n) = A238534(n).
Phi_7(n) = phi(n^7) = n^6*phi(n)= A239442(n).
Phi_8(n) = A239441(n).
Phi_9(n) = phi(n^9) = n^8*phi(n)= A239443(n).

with(numtheory); A239443:=n->phi(n^9); seq(A239443(n), n=1..100); # Wesley Ivan Hurt, Apr 01 2014

Mathematica
```
Table[EulerPhi[n^9], {n, 100}]
```

a(n) = n^8*eulerphi(n); \\ Michel Marcus, Mar 10 2018

Dirichlet g.f.: zeta(s - 9) / zeta(s - 8). The n-th term of the Dirichlet inverse is n^8 * A023900(n) = (-1)^omega(n) * a(n) / A003557(n), where omega = A001221. - Álvar Ibeas, Nov 24 2017
a(n) = n^8 * phi(n). - Altug Alkan, Mar 10 2018
Sum_{k=1..n} a(k) ~ 3*n^10 / (5*Pi^2). - Vaclav Kotesovec, Feb 02 2019
Sum_{n>=1} 1/a(n) = Product_{p prime} (1 + p/(p^10 - p^9 - p + 1)) = 1.00399107654133714629... - Amiram Eldar, Dec 06 2020

A238534 Number of solutions to gcd(u^2 + v^2 + w^2 + x^2 + y^2 + z^2, n) = 1 with u, v, w, x, y, z in [0,n-1].

Original entry on oeis.org

1, 32, 504, 2048, 12400, 16128, 101136, 131072, 367416, 396800, 1611720, 1032192, 4453488, 3236352, 6249600, 8388608, 22713088, 11757312, 44576280, 25395200, 50972544, 51575040, 141611184, 66060288, 193750000, 142511616, 267846264, 207126528, 574288624

Offset: 1

José María Grau Ribas, Feb 28 2014

Robert Israel, Table of n, a(n) for n = 1..2000 (first 100 terms from Giovanni Resta)
Catalina Calderón, Jose Maria Grau, A. Oller-Marcén, and László Tóth, Counting invertible sums of squares modulo n and a new generalization of Euler's totient function, Publicationes Mathematicae-Debrecen, Vol. 87 (1-2) (2015), pp. 133-145; arXiv preprint, arXiv:1403.7878 [math.NT], 2014.

Cf. A227499, A238533, A239441.

f:= proc(n) local i, j, k, S1, S2,  S4,  S6,G;
  G:= select(t -> igcd(t,n)=1, [$1..n-1]);
  S1:= Array(0..n-1);
  for i from 0 to n-1 do j:= i^2 mod n; S1[j]:= S1[j]+1; od;
  S2:= Array(0..n-1);
  for i from 0 to n-1 do
    for j from 0 to n-1 do
      k:= i^2 + j mod n;
      S2[k]:= S2[k]+S1[j];
  od od:
  S4:= Array(0..n-1);
  for i from 0 to n-1 do
    for j from 0 to n-1 do
      k:= i + j mod n;
      S4[k]:= S4[k]+S2[i]*S2[j];
  od od:
  S6:= Array(0..n-1);
  for i from 0 to n-1 do
    for j from 0 to n-1 do
      k:= i + j mod n;
      S6[k]:= S6[k]+S4[i]*S2[j];
  od od:
  add(S6[i],i=G);
end proc:
f(1):= 1:
map(f, [$1..100]); # Robert Israel, Mar 05 2018

g[n_, 6] := g[n, 6] = Sum[If[GCD[u^2+v^2+w^2+x^2+y^2+z^2, n] == 1, 1, 0], {u, n}, {v, n}, {w, n}, {x, n}, {y, n}, {z, n}]; Table[g[n, 6], {n, 1, 12}]
f[p_, e_] := (p - 1)*p^(6*e - 4)*(p^3 - (-1)^(3*(p - 1)/2)); f[2, e_] := 2^(6*e - 1); a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 30] (* Amiram Eldar, Sep 07 2023 *)

a(n)={my(p=lift(Mod(sum(i=0, n-1, x^(i^2%n)), x^n-1)^6)); sum(i=0, n-1, if(gcd(i,n)==1, polcoeff(p,i)))} \\ Andrew Howroyd, Aug 06 2018

a(n)={my(f=factor(n)); prod(i=1, #f~, my([p,e]=f[i,]); if(p==2, 2^(6*e-1), (p - 1)*p^(6*e - 4)*(p^3 - (-1)^(3*(p-1)/2))))} \\ Andrew Howroyd, Aug 07 2018

Multiplicative with a(2^e) = 2^(6*e-1), a(p^e) = (p - 1)*p^(6*e - 4)*(p^3 - (-1)^(3*(p-1)/2)) for odd prime p. - Andrew Howroyd, Aug 07 2018
From Amiram Eldar, Feb 13 2024: (Start)
Dirichlet g.f.: zeta(s-6) * (1 - 1/2^(s-5)) * Product_{p prime > 2} (1 - 1/p^(s-5) - (-1)^(3*(p-1)/2)*(p-1)/p^(s-2)).
Sum_{k=1..n} a(k) = c * n^7 + O(n^6 * log(n)), where c = (3/28) * Product_{p prime == 1 (mod 4)} (1 - 1/p^2 - 1/p^4 + 1/p^5) * Product_{p prime == 3 (mod 4)} (1 - 1/p^2 + 1/p^4 - 1/p^5) = 0.08756841635... (Calderón et al., 2015). (End)

a(16)-a(29) from Giovanni Resta, Mar 05 2014

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A079458 Number of Gaussian integers in a reduced system modulo n.

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A239442 a(n) = phi(n^7).

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A239443 a(n) = phi(n^9), where phi = A000010.

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A238534 Number of solutions to gcd(u^2 + v^2 + w^2 + x^2 + y^2 + z^2, n) = 1 with u, v, w, x, y, z in [0,n-1].

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