A239614 -id:A239614 - cp's OEIS frontend

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

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

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

cp's OEIS Frontend

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

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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).

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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).

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

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