A227499 -id:A227499 - cp's OEIS frontend

A000252 Number of invertible 2 X 2 matrices mod n.

1, 6, 48, 96, 480, 288, 2016, 1536, 3888, 2880, 13200, 4608, 26208, 12096, 23040, 24576, 78336, 23328, 123120, 46080, 96768, 79200, 267168, 73728, 300000, 157248, 314928, 193536, 682080, 138240, 892800, 393216, 633600, 470016, 967680, 373248, 1822176, 738720

Offset: 1

N. J. A. Sloane

For a prime p, a(p) = (p^2 - 1)*(p^2 - p) (this is the order of GL(2,p)). More generally a(n) is multiplicative: if the canonical factorization of n is the Product_{i=1..k} (p_i)^(e_i), then a(n) = Product_{i=1..k} (((p_i)^(2*e_i) - (p_i)^(2*e_i - 2)) * ((p_i)^(2*e_i) - (p_i)^(2*e_i - 1))). - Brian Wallace (wallacebrianedward(AT)yahoo.co.uk), Apr 05 2001, Dan Fux (dan.fux(AT)OpenGaia.com or danfux(AT)OpenGaia.com), Apr 18 2001
a(n) is the order of the automorphism group of the group C_n X C_n, where C_n is the cyclic group of order n. - Laszlo Toth, Dec 06 2011
Order of the group GL(2,Z_n). For n > 2, a(n) is divisible by 48. - Jianing Song, Jul 08 2018

T. D. Noe, Table of n, a(n) for n = 1..1000
Geoffrey Critzer, Combinatorics of Vector Spaces over Finite Fields, Master's thesis, Emporia State University, 2018.
C. J. Hillar and D. L. Rhea, Automorphisms of finite abelian groups, arXiv:math/0605185 [math.GR], 2006.
C. J. Hillar and D. L. Rhea, Automorphisms of finite abelian groups, Amer. Math. Monthly 114 (2007), no 10, 917-923.
J. Overbey, W. Traves and J. Wojdylo, On the Keyspace of the Hill Cipher, Cryptologia, Vol. 29 , Iss. 1, 2005.

The order of GL_2(K) for a finite field K is in sequence A059238.
Row n=2 of A316622.
Row sums of A316566.
Cf. A064767 (GL(3,Z_n)), A305186 (GL(4,Z_n)).
Cf. A000056 (SL(2,Z_n)), A011785 (SL(3,Z_n)), A011786 (SL(4,Z_n)).
Cf. A227499.

Table[n*EulerPhi[n]*Sum[d^2 MoebiusMu[n/d], {d, Divisors[n]}], {n, 21}] (* Jean-François Alcover, Apr 04 2011, after Vladeta Jovovic *)

a(n)=my(f=factor(n)[,1]); n^4*prod(i=1,#f, (1-1/f[i]^2)*(1-1/f[i])) \\ Charles R Greathouse IV, Feb 06 2017

from math import prod
from sympy import factorint
def A000252(n): return prod(p**((e<<2)-3)*(p*(p*(p-1)-1)+1) for p,e in factorint(n).items()) # Chai Wah Wu, Mar 04 2025

a(n) = n^4*Product_{primes p dividing n} (1 - 1/p^2)*(1 - 1/p) = n^4*Product_{primes p dividing n} p^(-3)*(p^2 - 1)*(p - 1). - Dan Fux (dan.fux(AT)OpenGaia.com or danfux(AT)OpenGaia.com), Apr 18 2001
Multiplicative with a(p^e) = (p - 1)^2*(p + 1)*p^(4e-3). - David W. Wilson, Aug 01 2001
a(n) = A000056(n)*phi(n), where phi is Euler totient function (cf. A000010). - Vladeta Jovovic, Oct 30 2001
Dirichlet g.f.: zeta(s - 4)*Product_{p prime} (1 - p^(1 - s)*(p^2 + p - 1)). - Álvar Ibeas, Nov 28 2017
a(n) = A227499(n) for odd n; (3/4)*A227499(n) for even n. - Jianing Song, Jul 08 2018
Sum_{k=1..n} a(k) ~ c * n^5 / 5, where c = A330523 = Product_{primes p} (1 - 1/p^2 - 1/p^3 + 1/p^4) = 0.5358961538283379998085... - Vaclav Kotesovec, Aug 20 2021
Sum_{n>=1} 1/a(n) = (Pi^8/3240) * Product_{p prime} (1 - 2/p^2 + 1/p^4 + 1/p^5 + 2/p^6 - 1/p^8) = 1.2059016071... . - Amiram Eldar, Dec 03 2022

More terms from David W. Wilson, Jul 21 2001

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

Original entry on oeis.org

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

A238533 Number of solutions to gcd(x^2 + y^2 + z^2 + t^2 + h^2, n) = 1 with x,y,z,t,h in [0,n-1].

Original entry on oeis.org

1, 16, 162, 512, 2500, 2592, 14406, 16384, 39366, 40000, 146410, 82944, 342732, 230496, 405000, 524288, 1336336, 629856, 2345778, 1280000, 2333772, 2342560, 6156502, 2654208, 7812500, 5483712, 9565938, 7375872, 19803868, 6480000, 27705630, 16777216, 23718420

Offset: 1

José María Grau Ribas, Feb 28 2014

Amiram Eldar, Table of n, a(n) for n = 1..10000
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. 9.

Cf. A079458, A227499, A238534.

Cf. n^k * a(n): A000010 (k=-4), A002618 (k=-3), A053191 (k=-2), A189393 (k=-1), A239442 (k=2), A239443 (k=4).

g[n_, 5] := g[n, 5] = Sum[If[GCD[x^2 + y^2 + z^2 + t^2 + h^2, n] == 1, 1, 0], {x, n}, {y, n}, {z, n}, {t, n}, {h, n}];Table[g[n,5] , {n, 1, 15}]
Table[n^4 * EulerPhi[n], {n, 1, 33}] (* Amiram Eldar, Dec 06 2020 *)

From Álvar Ibeas, Nov 24 2017: (Start)
a(n) = phi(n^5) = n^4 * phi(n), where phi=A000010.
Dirichlet g.f.: zeta(s - 5) / zeta(s - 4). The n-th term of the Dirichlet inverse is n^4 * A023900(n) = (-1)^omega(n) * a(n) / A003557(n), where omega = A001221.
(End)
Sum_{k=1..n} a(k) ~ n^6 / Pi^2. - Vaclav Kotesovec, Feb 02 2019
Sum_{n>=1} 1/a(n) = Product_{p prime} (1 + p/(p^6 - p^5 - p + 1)) = 1.07162935672651489627... - Amiram Eldar, Dec 06 2020

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

A239441 Number of invertible octonions over Z/nZ.

Original entry on oeis.org

1, 128, 4320, 32768, 312000, 552960, 4939200, 8388608, 28343520, 39936000, 194858400, 141557760, 752955840, 632217600, 1347840000, 2147483648, 6565340160, 3627970560, 16089567840, 10223616000, 21337344000, 24941875200, 74905892160, 36238786560, 121875000000, 96378347520

Offset: 1

José María Grau Ribas, Mar 18 2014

Number of octonions over Z/nZ with invertible norm; i.e., number of solutions of the equation gcd(x_1^2 + ... + x_8^2, n)=1 with 0 < x_i <= n.

Andrew Howroyd, Table of n, a(n) for n = 1..1000
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.

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).

fa=FactorInteger;lon[n_]:=Length[fa[n]];Phi[k_, n_] := Which[Mod[k, 2] == 1, n^(k - 1)*EulerPhi[n], Mod[k, 4] ==0, n^(k - 1)*EulerPhi[n]*Product[1 - 1/fa[2n][[i, 1]]^(k/2), {i, 2, lon[2 n]}],True, n^(k - 1)*EulerPhi[n]*Product[Which[ Mod[fa[ n][[i, 1]], 4] == 3 , 1 + 1/fa[ n][[i, 1]]^(k/2), Mod[fa[ n][[i, 1]], 4] == 1, 1 - 1/fa[ n][[i, 1]]^(k/2), True, 1], {i, 1, lon[ n]}]]; Table[Phi[8,n],{n,1,100}]
f[p_, e_] := (p-1)*p^(8*e-1) * If[p == 2, 1, 1 - 1/p^4]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 30] (* Amiram Eldar, Feb 13 2024 *)

a(n)={my(p=lift(Mod(sum(i=0, n-1, x^(i^2%n)), x^n-1)^8)); 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^(8*e-1), (p - 1)*p^(8*e - 5)*(p^4 - 1)))} \\ Andrew Howroyd, Aug 06 2018

Multiplicative with a(2^e) = 2^(8*e-1), a(p^e) = (p - 1)*p^(8*e - 5)*(p^4 - 1) for odd prime p. - Andrew Howroyd, Aug 06 2018
Sum_{k=1..n} a(k) ~ c * n^9, where c = (16/141) * Product_{p prime} (1 - 1/p^2 - 1/p^5 + 1/p^6) = 0.06731687367... . - Amiram Eldar, Nov 30 2022
From Amiram Eldar, Feb 13 2024: (Start)
Dirichlet g.f.: zeta(s-8) * (1 - 1/2^(s-7)) * Product_{p prime > 2} (1 - 1/p^(s-7) - (p-1)/p^(s-3)).
Sum_{n>=1} 1/a(n) = (257*Pi^14/1312151400) * Product_{p prime} (1 - 1/p^2 - 1/p^4 + 1/p^6 + 1/p^9 + 1/p^10 + 1/p^12 - 1/p^14) = 1.00807991170717322545... . (End)

A239611 a(n) = Sum_{0 < x,y <= n and gcd(x^2 + y^2, n)=1} gcd(x^2 + y^2 - 1, n).

Original entry on oeis.org

1, 4, 16, 32, 32, 64, 96, 192, 216, 128, 240, 512, 288, 384, 512, 1024, 512, 864, 720, 1024, 1536, 960, 1056, 3072, 1200, 1152, 2592, 3072, 1568, 2048, 1920, 5120, 3840, 2048, 3072, 6912, 2592, 2880, 4608, 6144, 3200, 6144, 3696, 7680, 6912, 4224, 4416

Offset: 1

José María Grau Ribas, Jun 24 2014

Related to Menon's identity. See Conclusions and further work section of the arXiv file linked.

Multiplicative by the Chinese remainder theorem since gcd(x, m*n) = gcd(x, m)*gcd(x, n) for gcd(m, n) = 1. - Andrew Howroyd, Aug 07 2018

Antti Karttunen, Table of n, a(n) for n = 1..2101
C. Calderón, J. M. Grau, A. Oller-Marcen, L. Toth, Counting invertible sums of squares modulo n and a new generalization of Euler totient function, arXiv:1403.7878 [math.NT], 2014.

Cf. A239612, A239613, A239614, A239615, A079458, A053191, A227499, A244342.

g2[n_] := Sum[If[GCD[x^2 + y^2, n] == 1, GCD[x^2 + y^2 - 1, n], 0], {x, 1, n}, {y, 1, n}]; Array[g2,100]

a(n) = {s = 0; for (x=1, n, for (y=1, n, if (gcd(x^2+y^2,n) == 1, s += gcd(x^2+y^2-1,n)););); s;} \\ Michel Marcus, Jun 29 2014

A239612 a(n) = Sum_{0 < x,y,z <= n and gcd(x^2 + y^2 + z^2, n)=1} gcd(x^2 + y^2 + z^2 - 1, n).

Original entry on oeis.org

1, 8, 30, 112, 220, 240, 546, 1280, 1134, 1760, 2310, 3360, 4212, 4368, 6600, 13312, 9520, 9072, 12654, 24640, 16380, 18480, 22770, 38400, 42500, 33696, 39366, 61152, 47908, 52800, 56730, 131072, 69300, 76160, 120120, 127008, 99900, 101232, 126360, 281600

Offset: 1

José María Grau Ribas, Jun 25 2014

Related to Menon's identity. See Conclusions and further work section of the arXiv file linked.

Andrew Howroyd, Table of n, a(n) for n = 1..1000
C. Calderón, J. M. Grau, A. Oller-Marcen, L. Toth, Counting invertible sums of squares modulo n and a new generalization of Euler totient function, arXiv:1403.7878 [math.NT], 2014.

Cf. A239611, A239613, A239614, A239615, A079458, A053191, A227499.

g3[n_] := Sum[If[GCD[x^2 + y^2 + z^2, n] == 1, GCD[x^2 + y^2 + z^2 - 1, n], 0],{x, 1, n},{y, 1, n},{z,1,n}]; Array[g3,100]

a(n) = {s = 0; for (x=1, n, for (y=1, n, for (z=1, n, if (gcd(x^2+y^2+z^2,n) == 1, s += gcd(x^2+y^2+z^2-1,n));););); s;} \\ Michel Marcus, Jun 29 2014

a(n)={my(p=lift(Mod(sum(i=0, n-1, x^(i^2%n)), x^n-1)^3)); sum(i=0, n-1, if(gcd(i,n)==1, polcoeff(p,i)*gcd((i-1)%n,n)))} \\ Andrew Howroyd, Aug 07 2018

Keyword:mult added by Andrew Howroyd, Aug 07 2018

A239613 a(n) = Sum_{0 < x,y,z,t <= n and gcd(x^2 + y^2 + z^2 + t^2, n)=1} gcd(x^2 + y^2 + z^2 + t^2 - 1, n).

Original entry on oeis.org

1, 16, 96, 384, 960, 1536, 4032, 8192, 11664, 15360, 26400, 36864, 52416, 64512, 92160, 163840, 156672, 186624, 246240, 368640, 387072, 422400, 534336, 786432, 900000, 838656, 1259712, 1548288, 1364160, 1474560, 1785600, 3145728

Offset: 1

José María Grau Ribas, Jun 25 2014

Related to Menon's identity. See Conclusions and further work section of the arXiv file linked.

Andrew Howroyd, Table of n, a(n) for n = 1..1000
C. Calderón, J. M. Grau, A. Oller-Marcen, L. Toth, Counting invertible sums of squares modulo n and a new generalization of Euler totient function, arXiv:1403.7878 [math.NT], 2014.

Cf. A239611, A239612, A239614, A239615, A079458, A053191, A227499.

g4[n_] := Sum[If[GCD[x^2 + y^2+ z^2+ t^2, n] == 1, GCD[x^2 + y^2+ z^2+ t^2 - 1, n], 0], {x, 1, n}, {y, 1, n},{z,1,n},{t,1,n}]; Array[g4,100]

a(n) = {s = 0; for (x=1, n, for (y=1, n, for (z=1, n, for (t=1, n, if (gcd(x^2+y^2+z^2+t^2,n) == 1, s += gcd(x^2+y^2+z^2+t^2-1,n)););););); s;} \\ Michel Marcus, Jun 29 2014

a(n)={my(p=lift(Mod(sum(i=0, n-1, x^(i^2%n)), x^n-1)^4)); sum(i=0, n-1, if(gcd(i,n)==1, polcoeff(p,i)*gcd((i-1)%n,n)))} \\ Andrew Howroyd, Aug 07 2018

Keyword:mult added by Andrew Howroyd, Aug 07 2018

cp's OEIS Frontend

A000252 Number of invertible 2 X 2 matrices mod n.

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

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A238533 Number of solutions to gcd(x^2 + y^2 + z^2 + t^2 + h^2, n) = 1 with x,y,z,t,h in [0,n-1].

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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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A239441 Number of invertible octonions over Z/nZ.

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