A053191 -id:A053191 - cp's OEIS frontend

A189393 a(n) = phi(n^4).

1, 8, 54, 128, 500, 432, 2058, 2048, 4374, 4000, 13310, 6912, 26364, 16464, 27000, 32768, 78608, 34992, 123462, 64000, 111132, 106480, 267674, 110592, 312500, 210912, 354294, 263424, 682892, 216000, 893730, 524288, 718740, 628864, 1029000, 559872

Offset: 1

Vincenzo Librandi, Apr 21 2011

Vincenzo Librandi and T. D. Noe, Table of n, a(n) for n = 1..1000 (terms 1..680 from Vincenzo Librandi)

Cf. A002618 (phi(n^2)), A053191 (phi(n^3)), A238533 (phi(n^5)), A239442 (phi(n^7)), A239443 (phi(n^9)).

Magma
```
[ n^3*EulerPhi(n) : n in [1..100] ]
```

EulerPhi[Range[100]^4] (* T. D. Noe, Dec 27 2011 *)

vector(66,n,n^3*eulerphi(n))  /* Joerg Arndt, Apr 22 2011 */

a(n) = n^3*phi(n).
Dirichlet g.f.: zeta(s - 4) / zeta(s - 3). The n-th term of the Dirichlet inverse is n^3 * A023900(n) = (-1)^omega(n) * a(n) / A003557(n), where omega=A001221. - Álvar Ibeas, Nov 24 2017
Sum_{k=1..n} a(k) ~ 6*n^5 / (5*Pi^2). - Vaclav Kotesovec, Feb 02 2019
Sum_{n>=1} 1/a(n) = Product_{p prime} (1 + p/(p^5 - p^4 - p + 1)) = 1.15762316629211803144... - Amiram Eldar, Dec 06 2020

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

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

Original entry on oeis.org

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

A074466 a(n) = gcd(n^3, sigma(n^3), phi(n^3)).

Original entry on oeis.org

1, 1, 1, 1, 1, 24, 1, 1, 1, 20, 1, 8, 1, 8, 15, 1, 1, 3, 1, 4, 1, 8, 1, 24, 1, 4, 1, 16, 1, 1800, 1, 1, 3, 4, 25, 1, 1, 8, 1, 4, 1, 24, 1, 8, 3, 8, 1, 8, 1, 5, 9, 4, 1, 12, 1, 16, 1, 4, 1, 480, 1, 8, 1, 1, 65, 72, 1, 4, 3, 200, 1, 3, 1, 4, 5, 8, 1, 24, 1, 4, 1, 4, 1, 64, 5, 8, 3, 88, 1, 180, 7, 16, 1

Offset: 1

Labos Elemer, Aug 23 2002

nonn

			n=10: gcd[1000,2340,400] = 20 = a(10).

Antti Karttunen, Table of n, a(n) for n = 1..65537

Cf. A053191, A074389, A074465, A175926.

[Gcd(n^3, Gcd(SumOfDivisors(n^3), EulerPhi(n^3))): n in [1..100]]; // Vincenzo Librandi, Sep 20 2018

Table[Apply[GCD, {w^3, DivisorSigma[1, w^3], EulerPhi[w^3]}], {w, 1, 128}]
GCD[#,DivisorSigma[1,#],EulerPhi[#]]&/@(Range[100]^3) (* Harvey P. Dale, Nov 03 2024 *)

A074466(n) = gcd([n^3, sigma(n^3), eulerphi(n^3)]); \\ Antti Karttunen, Sep 07 2018

a(n) = A074389(n^3).

A306411 a(n) = phi(n^6) = n^5*phi(n).

Original entry on oeis.org

1, 32, 486, 2048, 12500, 15552, 100842, 131072, 354294, 400000, 1610510, 995328, 4455516, 3226944, 6075000, 8388608, 22717712, 11337408, 44569782, 25600000, 49009212, 51536320, 141599546, 63700992, 195312500, 142576512, 258280326, 206524416, 574312172, 194400000, 858874530, 536870912, 782707860, 726966784

Offset: 1

Jianing Song, Feb 13 2019

The number of elements of the wreath product of C_n and S_6 with cycle partition equal to (6*n) is equal to 5!*a(n), where C_n is the cyclic group of order n, S_6 the symmetric group on 6 elements. - Josaphat Baolahy, Mar 13 2024

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

Eulerphi(n^e): A000010 (e=1), A002618 (e=2), A053191 (e=3), A189393 (e=4), A238533 (e=5), this sequence (e=6), A239442 (e=7), A306412 (e=8), A239443 (e=9).

Array[EulerPhi[#] #^5 &, 34] (* Michael De Vlieger, Feb 17 2019 *)

PARI
```
a(n) = n^5 * eulerphi(n)
```

Multiplicative with a(p^e) = (p - 1)*p^(6*e-1).
Dirichlet g.f.: zeta(s - 6) / zeta(s - 5).
Sum_{k=1..n} a(k) ~ 6*n^7 / (7*Pi^2). See A239443 for a more general formula.
Sum_{k>=1} 1/a(k) = Product_{primes p} (1 + p/(p^7 - p^6 - p + 1)) = 1.03396580456393429553879930771676667947490034699829164744357501993310897305... - Vaclav Kotesovec, Sep 20 2020

A306412 a(n) = phi(n^8) = n^7*phi(n).

Original entry on oeis.org

1, 128, 4374, 32768, 312500, 559872, 4941258, 8388608, 28697814, 40000000, 194871710, 143327232, 752982204, 632481024, 1366875000, 2147483648, 6565418768, 3673320192, 16089691302, 10240000000, 21613062492, 24943578880, 74906159834, 36691771392, 122070312500

Offset: 1

Jianing Song, Feb 13 2019

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

Eulerphi(n^e): A000010 (e=1), A002618 (e=2), A053191 (e=3), A189393 (e=4), A238533 (e=5), A306411 (e=6), A239442 (e=7), this sequence (e=8), A239443 (e=9).

Table[n^7*EulerPhi[n], {n, 1, 25}] (* Amiram Eldar, Dec 06 2020 *)

PARI
```
a(n) = n^7 * eulerphi(n)
```

Multiplicative with a(p^e) = (p - 1)*p^(8*e-1).
Dirichlet g.f.: zeta(s - 8) / zeta(s - 7).
Sum_{k=1..n} a(k) ~ 2*n^9 / (3*Pi^2). See A239443 for a more general formula.
Sum_{n>=1} 1/a(n) = Product_{p prime} (1 + p/(p^9 - p^8 - p + 1)) = 1.00807702579309679541... - Amiram Eldar, Dec 06 2020

A053198 Totients of consecutive pure powers of primes.

Original entry on oeis.org

2, 4, 6, 8, 20, 18, 16, 42, 32, 54, 110, 100, 64, 156, 162, 128, 272, 294, 342, 256, 506, 500, 486, 812, 930, 512, 1210, 1332, 1640, 1806, 1024, 1458, 2028, 2162, 2058, 2756, 2500, 3422, 3660, 2048, 4422, 4624, 4970, 5256, 6162, 4374, 6498, 6806, 7832, 4096

Offset: 1

Labos Elemer, Mar 03 2000

nonn

Totients of prime powers are prime powers only for powers of 2.

			The 10th pure power of prime (but not a prime) is 81, so a(10) = EulerPhi(81) = 54.

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

Cf. A000010, A051953, A001248, A002618, A036689, A053650, A053191, A053192, A086242.

EulerPhi[Select[Range[2^13], CompositeQ[#] && PrimePowerQ[#] &]] (* Amiram Eldar, Dec 21 2020 *)

a(n) = A000010(A025475(n+1)).
Numbers of the form phi(p^k) = (p-1)*p^(k-1), where p is prime and k > 1.
Sum_{n>=1} 1/a(n) = Sum_{p prime} 1/(p-1)^2 = A086242 = 1.3750649947... - Amiram Eldar, Dec 21 2020

cp's OEIS Frontend

A189393 a(n) = phi(n^4).

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A239441 Number of invertible octonions over Z/nZ.

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

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A074466 a(n) = gcd(n^3, sigma(n^3), phi(n^3)).

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