cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-6 of 6 results.

A372964 a(n) = Sum_{1 <= x_1, x_2, x_3, x_4 <= n} ( n/gcd(x_1, x_2, x_3, x_4, n) )^3.

Original entry on oeis.org

1, 121, 2161, 15481, 78001, 261481, 823201, 1981561, 4726081, 9438121, 19485841, 33454441, 62746321, 99607321, 168560161, 253639801, 410333761, 571855801, 893864881, 1207533481, 1778937361, 2357786761, 3404813281, 4282153321, 6093828001, 7592304841, 10335939121
Offset: 1

Views

Author

Seiichi Manyama, May 18 2024

Keywords

Links

Crossrefs

Cf. A068970, A084220, A372950.
Cf. A008683, A013955.
Cf. A013663, A013666.
Cf. A350156, A372962.

Programs

  • Mathematica
    f[p_, e_] := (p^(7*e+7) - p^(7*e+3) + p^3 - 1)/(p^7-1); a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, May 21 2024 *)
  • PARI
    a(n) = sumdiv(n, d, moebius(n/d)*(n/d)^3*sigma(d, 7));

Formula

a(n) = Sum_{d|n} mu(n/d) * (n/d)^3 * sigma_7(d).
From Amiram Eldar, May 21 2024: (Start)
Multiplicative with a(p^e) = (p^(7*e+7) - p^(7*e+3) + p^3 - 1)/(p^7-1).
Dirichlet g.f.: zeta(s)*zeta(s-7)/zeta(s-3).
Sum_{k=1..n} a(k) ~ c * n^8 / 8, where c = zeta(8)/zeta(5) = 0.968319491... . (End)
a(n) = Sum_{d|n} phi(n/d) * (n/d)^6 * sigma_6(d^2)/sigma_3(d^2). - Seiichi Manyama, May 24 2024
a(n) = Sum_{1 <= x_1, x_2, x_3, x_4 <= n} ( gcd(x_1, n)/gcd(x_1, x_2, x_3, x_4, n) )^4. - Seiichi Manyama, May 25 2024

A321349 a(n) = Sum_{d|n} phi(d^n), where phi() is the Euler totient function (A000010).

Original entry on oeis.org

1, 3, 19, 137, 2501, 16071, 705895, 8421505, 258293449, 4007813013, 259374246011, 2972767821815, 279577021469773, 4762869973595499, 233543432626753439, 9223512776490647553, 778579070010669895697, 13115569455375954492093, 1874292305362402347591139
Offset: 1

Views

Author

Ilya Gutkovskiy, Nov 06 2018

Keywords

Links

Crossrefs

Cf. A000010, A057660, A068963, A068970, A226459, A226561.

Programs

  • Mathematica
    Table[Sum[EulerPhi[d^n], {d, Divisors[n]}], {n, 19}]
nmax = 19; Rest[CoefficientList[Series[Sum[k^(k - 1) EulerPhi[k] x^k/(1 - (k x)^k), {k, 1, nmax}], {x, 0, nmax}], x]]
Table[Sum[(n/GCD[n, k])^(n - 1), {k, n}], {n, 19}]
  • PARI
    a(n) = sumdiv(n, d, eulerphi(d^n)); \\ Michel Marcus, Nov 06 2018

Formula

G.f.: Sum_{k>=1} k^(k-1)*phi(k)*x^k/(1 - (k*x)^k).
a(n) = Sum_{d|n} d^(n-1)*phi(d).
a(n) = Sum_{k=1..n} (n/gcd(n,k))^(n-1).
From Richard L. Ollerton, May 08 2021: (Start)
a(n) = Sum_{k=1..n} phi(gcd(n,k)^n)/phi(n/gcd(n,k)).
a(n) = Sum_{k=1..n} gcd(n,k)^(n-1)*phi(gcd(n,k))/phi(n/gcd(n,k)). (End)

A371491 a(n) = Sum_{1 <= x_1, x_2, x_3, x_4, x_5 <= n} ( n/gcd(x_1, x_2, x_3, x_4, x_5, n) )^3.

Original entry on oeis.org

1, 249, 6535, 63737, 390501, 1627215, 5764459, 16316665, 42876109, 97234749, 214357551, 416521295, 815728525, 1435350291, 2551924035, 4177066233, 6975752529, 10676151141, 16983556183, 24889362237, 37670739565, 53375030199, 78310973115, 106629405775
Offset: 1

Views

Author

Seiichi Manyama, May 24 2024

Keywords

Links

Crossrefs

Cf. A068970, A084220, A372950, A372964.
Cf. A372952, A372963.
Cf. A372966.
Cf. A013664, A013667.

Programs

  • Mathematica
    f[p_, e_] := (p^(8*e + 8) - p^(8*e + 3) + p^3 - 1)/(p^8 - 1); a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 24] (* Amiram Eldar, May 24 2024 *)
  • PARI
    a(n) = sumdiv(n, d, moebius(n/d)*(n/d)^3*sigma(d, 8));
  • PARI
    a(n) = sumdiv(n,d, eulerphi(n/d)*(n/d)^3*sigma(d^2, 8)/sigma(d^2, 4));

Formula

a(n) = Sum_{d|n} mu(n/d) * (n/d)^3 * sigma_8(d).
a(n) = Sum_{d|n} phi(n/d) * (n/d)^3 * sigma_8(d^2)/sigma_4(d^2).
From Amiram Eldar, May 24 2024: (Start)
Multiplicative with a(p^e) = (p^(8*e+8) - p^(8*e+3) + p^3 - 1)/(p^8-1).
Dirichlet g.f.: zeta(s)*zeta(s-8)/zeta(s-3).
Sum_{k=1..n} a(k) ~ c * n^9 / 9, where c = zeta(9)/zeta(6) = 0.984926747... . (End)
a(n) = Sum_{1 <= x_1, x_2, x_3, x_4, x_5 <= n} ( gcd(x_1, x_2, n)/gcd(x_1, x_2, x_3, x_4, x_5, n) )^5. - Seiichi Manyama, May 25 2024

A372950 a(n) = Sum_{1 <= x_1, x_2 <= n} ( n/gcd(x_1, x_2, n) )^3.

Original entry on oeis.org

1, 25, 217, 793, 3001, 5425, 16465, 25369, 52705, 75025, 159721, 172081, 369097, 411625, 651217, 811801, 1414945, 1317625, 2469241, 2379793, 3572905, 3993025, 6424177, 5505073, 9378001, 9227425, 12807289, 13056745, 20486761, 16280425, 28599361, 25977625, 34659457
Offset: 1

Views

Author

Seiichi Manyama, May 18 2024

Keywords

Links

Crossrefs

Cf. A084218, A350156.
Cf. A068970, A084220, A372964.
Cf. A001160, A008683.
Cf. A002117, A013664, A347328.

Programs

  • Mathematica
    f[p_, e_] := (p^(5*e+5) - p^(5*e+3) + p^3 - 1)/(p^5-1); a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, May 21 2024 *)
  • PARI
    a(n) = sumdiv(n, d, moebius(n/d)*(n/d)^3*sigma(d, 5));

Formula

a(n) = Sum_{d|n} mu(n/d) * (n/d)^3 * sigma_5(d).
From Amiram Eldar, May 21 2024: (Start)
Multiplicative with a(p^e) = (p^(5*e+5) - p^(5*e+3) + p^3 - 1)/(p^5-1).
Dirichlet g.f.: zeta(s)*zeta(s-5)/zeta(s-3).
Sum_{k=1..n} a(k) ~ c * n^6 / 6, where c = zeta(6)/zeta(3) = 0.846335... (A347328). (End)
Dirichlet convolution of A334659 and A001160. - R. J. Mathar, Jul 14 2025

A372965 a(n) = Sum_{k = 1..n} ( n/gcd(k, n) )^4.

Original entry on oeis.org

1, 17, 163, 529, 2501, 2771, 14407, 16913, 39529, 42517, 146411, 86227, 342733, 244919, 407663, 541201, 1336337, 671993, 2345779, 1323029, 2348341, 2488987, 6156503, 2756819, 7815001, 5826461, 9605467, 7621303, 19803869, 6930271, 27705631, 17318417, 23864993
Offset: 1

Views

Author

Seiichi Manyama, May 18 2024

Keywords

Links

Crossrefs

Cf. A057660, A068963, A068970.
Cf. A001160, A008683.
Cf. A372966.
Cf. A013661, A013664, A157292.

Programs

  • Mathematica
    f[p_, e_] := (p^(5*e+5) - p^(5*e+4) + p^4 - 1)/(p^5-1); a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, May 21 2024 *)
  • PARI
    a(n) = sumdiv(n, d, moebius(n/d)*(n/d)^4*sigma(d, 5));
  • PARI
    a(n) = sumdiv(n, d, eulerphi(d^5));

Formula

a(n) = Sum_{d|n} mu(n/d) * (n/d)^4 * sigma_5(d).
a(n) = Sum_{d|n} d^(5-m) * phi(d^m) for m > 0.
G.f.: Sum_{k>=1} k^(5-m) * phi(k^m) * x^k/(1 - x^k) for m > 0.
From Amiram Eldar, May 21 2024: (Start)
Multiplicative with a(p^e) = (p^(5*e+5) - p^(5*e+4) + p^4 - 1)/(p^5-1).
Dirichlet g.f.: zeta(s)*zeta(s-5)/zeta(s-4).
Sum_{k=1..n} a(k) ~ c * n^6 / 6, where c = zeta(6)/zeta(2) = 2*Pi^4/315 = 0.6184704192... (1/A157292). (End)

A344526 a(n) = Sum_{k=1..n} k^3 * phi(k).

Original entry on oeis.org

1, 9, 63, 191, 691, 1123, 3181, 5229, 9603, 13603, 26913, 33825, 60189, 76653, 103653, 136421, 215029, 250021, 373483, 437483, 548615, 655095, 922769, 1033361, 1345861, 1556773, 1911067, 2174491, 2857383, 3073383, 3967113, 4491401, 5210141, 5839005, 6868005, 7427877, 9251385
Offset: 1

Views

Author

Seiichi Manyama, May 22 2021

Keywords

Crossrefs

Cf. A000010, A011755, A068970, A319087, A344522.

Programs

  • Mathematica
    a[n_] := Sum[k^3 * EulerPhi[k], {k, 1, n}]; Array[a, 40] (* Amiram Eldar, May 22 2021 *)
Accumulate[Table[k^3*EulerPhi[k], {k, 1, 40}]] (* Vaclav Kotesovec, May 22 2021 *)
  • PARI
    a(n) = sum(k=1, n, k^3*eulerphi(k));

Formula

a(n) ~ 6*n^5 / (5*Pi^2). - Vaclav Kotesovec, May 22 2021
Showing 1-6 of 6 results.

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