A227499 -id:A227499 - cp's OEIS frontend

A239614 a(n) = A239611(n) / A079458(n).

1, 2, 2, 4, 2, 4, 2, 6, 3, 4, 2, 8, 2, 4, 4, 8, 2, 6, 2, 8, 4, 4, 2, 12, 3, 4, 4, 8, 2, 8, 2, 10, 4, 4, 4, 12, 2, 4, 4, 12, 2, 8, 2, 8, 6, 4, 2, 16, 3, 6, 4, 8, 2, 8, 4, 12, 4, 4, 2, 16, 2, 4, 6, 12, 4, 8, 2, 8, 4, 8, 2, 18, 2, 4, 6, 8, 4, 8, 2, 16, 5, 4, 2, 16, 4, 4, 4, 12, 2, 12, 4, 8, 4, 4, 4, 20, 2, 6, 6, 12, 2, 8, 2, 12, 8

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.

Multiplicative because both A239611 and A079458 are. - Andrew Howroyd, Aug 07 2018

Antti Karttunen, Table of n, a(n) for n = 1..2101 (computed from the b-files of A079458 and A239611)
C. Calderón, J. M. Grau, A. Oller-Marcen, and Laszlo 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, A239613, A239615, A079458, A053191, A227499.

a239611[n_] := Sum[If[GCD[x^2 + y^2, n] == 1, GCD[x^2 + y^2 - 1, n], 0], {x, 1, n}, {y, 1, n}];
a079458[n_] := Product[{p, e} = pe; Which[p==2, 2^(2e-1), Mod[p, 4]==3, (p^2-1)p^(2e-2), Mod[p, 4]==1, (p-1)^2 p^(2e-2)], {pe, FactorInteger[n]}];
a[1] = 1; a[n_] := a239611[n]/a079458[n];
Array[a, 105] (* Jean-François Alcover, Dec 04 2018 *)

Conjectures from Ridouane Oudra, Jul 22 2024: (Start)
a(n) = A010710(n)*tau(n) - 2*tau(2n) ;
a(2*n) = 2*tau(n) ;
a(2*n+1) = tau(2*n+1). (End)

More terms from Antti Karttunen, Sep 23 2017

A239615 a(n) = n * A239612(n) / A053191(n).

Original entry on oeis.org

1, 4, 5, 14, 11, 20, 13, 40, 21, 44, 21, 70, 27, 52, 55, 104, 35, 84, 37, 154, 65, 84, 45, 200, 85, 108, 81, 182, 59, 220, 61, 256, 105, 140, 143, 294, 75, 148, 135, 440, 83, 260, 85, 294, 231, 180, 93, 520, 133, 340, 175, 378, 107, 324, 231, 520, 185, 236

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.

Multiplicative because both A239612 and A053191 are. - Andrew Howroyd, Aug 07 2018

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, A239613, A239614, A079458, A053191, A227499.

a[n_] := Sum[Boole[GCD[x^2 + y^2 + z^2, n] == 1] GCD[x^2 + y^2 + z^2 - 1, n], {x, 1, n}, {y, 1, n}, {z, 1, n}]/(n EulerPhi[n]);
Array[a, 60] (* Jean-François Alcover, Nov 22 2018 *)

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)))/(n * eulerphi(n))} \\ Andrew Howroyd, Aug 07 2018

A227477 Exponent of the group of Lipschitz quaternions in a reduced system modulo n.

Original entry on oeis.org

1, 2, 24, 4, 120, 24, 336, 8, 72, 120

Offset: 1

José María Grau Ribas, Jul 13 2013

Cf. A227499, A002322.

cuaternios[n_] := Flatten[Table[{{a,-b,d ,-c},{b,a,-c,-d},{-d,c,a,-b},{c,d,b,a}},{a,n},{b,n},{c,n},{d,n}],3]; A227499[n_]:=Length@Select[cuaternios[n],GCD[Det[#],n]== 1 &]; cuater[n_] := Select[cuaternios[n], GCD[Det[#], n] == 1 &]; exp[1]=1; expo[M_,n_]:= Min@Select[Divisors@A227499[n],Mod[MatrixPower[M, #],n] == IdentityMatrix[4]&];a[n_] := lcm@Table[expo[cuater[n][[i]], n], {i, A227499[n]}]; lcm[lis_] := {aux = 1; Do[aux = LCM[aux, lis[[i]]], {i, 1, Length[lis]}]; aux}[[1]]; Table[a[n], {n,2,10}]

A227867 Number of Lipschitz quaternions X such that X^2 == 1 (mod n).

Original entry on oeis.org

1, 8, 14, 32, 32, 112, 58, 32, 110, 256, 134, 448, 184, 464, 448, 32, 308, 880, 382, 1024, 812, 1072, 554, 448, 752, 1472, 974, 1856, 872, 3584, 994, 32, 1876, 2464, 1856, 3520, 1408, 3056, 2576, 1024, 1724, 6496, 1894, 4288, 3520, 4432, 2258, 448, 2746, 6016, 4312, 5888

Offset: 1

José María Grau Ribas, Nov 02 2013

A quaternion q = a + bi + cj + dk is congruent to 1 (mod n) iff a == 1 (mod n) and b == c == d == 0 (mod n).

Wikipedia, Hurwitz quaternion.

Cf. A227477, A227499, A227628.

cuaternios[n_] := Flatten[Table[{{a, -b, d, -c}, {b, a, -c, -d}, {-d, c, a, -b}, {c, d, b, a}}, {a, n}, {b, n}, {c, n}, {d, n}], 3]; invo[n_] := invo[n] = Length@Select[cuaternios[n], Mod[#.# - IdentityMatrix[4],n] == 0*# &]; Table[invo[n], {n, 1, 25}]

More terms from Amiram Eldar, May 06 2024

A227628 Number of Lipschitz quaternions X such that X^2 == X (mod n).

Original entry on oeis.org

1, 2, 14, 2, 32, 28, 58, 2, 110, 64, 134, 28, 184, 116, 448, 2, 308, 220, 382, 64, 812, 268, 554, 28, 752, 368, 974, 116, 872, 896, 994, 2, 1876, 616, 1856, 220, 1408, 764, 2576, 64, 1724, 1624, 1894, 268, 3520, 1108, 2258, 28, 2746, 1504

Offset: 1

José María Grau Ribas, Jul 18 2013

C. J. Miguel and R. Serodio, On the Structure of Quaternion Rings over Zp, International Journal of Algebra, Vol. 5, 2011, no. 27, pp. 1313-1325.

Cf. A000118, A227477, A227499.

cuaternios[n_] := Flatten[Table[{{ a, -b, d, -c}, {b, a, -c, -d}, {-d, c, a, -b}, {c, d, b, a}}, {a, n}, {b, n}, {c, n}, {d, n}], 3]; cuater[n_] := Length@Select[cuaternios[n], Mod[#.# - #, n] == 0*# &]; Table[cuater[n],{n,1,100}]

cp's OEIS Frontend

A239614 a(n) = A239611(n) / A079458(n).

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A239615 a(n) = n * A239612(n) / A053191(n).

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A227477 Exponent of the group of Lipschitz quaternions in a reduced system modulo n.

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Mathematica

A227867 Number of Lipschitz quaternions X such that X^2 == 1 (mod n).

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Author

Keywords

Comments

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Programs

Mathematica

Extensions

A227628 Number of Lipschitz quaternions X such that X^2 == X (mod n).

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Author

Keywords

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Programs

Mathematica