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 1-5 of 5 results.

A344522 a(n) = Sum_{1 <= i, j, k <= n} gcd(i,j,k).

Original entry on oeis.org

1, 9, 30, 76, 141, 267, 400, 624, 885, 1249, 1590, 2208, 2689, 3411, 4248, 5248, 6081, 7485, 8530, 10248, 11889, 13687, 15228, 17988, 20053, 22569, 25242, 28588, 31053, 35463, 38284, 42540, 46581, 50893, 55362, 61824, 65857, 71247, 76884, 84388, 89349, 97881, 103342
Offset: 1

Views

Author

Seiichi Manyama, May 22 2021

Keywords

Links

Crossrefs

Column k=3 of A344479.
Cf. A018806, A071778, A343497, A344132, A344521, A344523, A344524, A344525, A344526.

Programs

  • Mathematica
    a[n_] := Sum[EulerPhi[k] * Quotient[n, k]^3, {k, 1, n}]; Array[a, 50] (* Amiram Eldar, May 22 2021 *)
  • PARI
    a(n) = sum(i=1, n, sum(j=1, n, sum(k=1, n, gcd([i, j, k]))));
  • PARI
    a(n) = sum(k=1, n, eulerphi(k)*(n\k)^3);
  • PARI
    my(N=66, x='x+O('x^N)); Vec(sum(k=1, N, eulerphi(k)*x^k*(1+4*x^k+x^(2*k))/(1-x^k)^3)/(1-x))

Formula

a(n) = Sum_{k=1..n} phi(k) * floor(n/k)^3.
G.f.: (1/(1 - x)) * Sum_{k >= 1} phi(k) * x^k * (1 + 4*x^k + x^(2*k))/(1 - x^k)^3.
a(n) ~ Pi^2 * n^3 / (6*zeta(3)). - Vaclav Kotesovec, May 23 2021

A343499 a(n) = Sum_{k=1..n} gcd(k, n)^5.

Original entry on oeis.org

1, 33, 245, 1058, 3129, 8085, 16813, 33860, 59541, 103257, 161061, 259210, 371305, 554829, 766605, 1083528, 1419873, 1964853, 2476117, 3310482, 4119185, 5315013, 6436365, 8295700, 9778145, 12253065, 14468481, 17788154, 20511177, 25297965, 28629181, 34672912, 39459945, 46855809
Offset: 1

Views

Author

Seiichi Manyama, Apr 17 2021

Keywords

Links

Crossrefs

Column 5 of A343510.
Cf. A000010, A001159 (sigma_4(n)), A018804, A054612, A059378, A069097, A332517, A342423, A342433, A343497, A343498, A344524.

Programs

  • Magma
    A343499:= func< n | (&+[d^5*EulerPhi(Floor(n/d)): d in Divisors(n)]) >;
[A343499(n): n in [1..50]]; // G. C. Greubel, Jun 24 2024
  • Mathematica
    a[n_] := Sum[GCD[k, n]^5, {k, 1, n}]; Array[a, 50] (* Amiram Eldar, Apr 18 2021 *)
f[p_, e_] := p^(e-1)*(p^(4*e+5) - p^(4*e) - p + 1)/(p^4-1); a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 50] (* Amiram Eldar, Nov 22 2022 *)
  • PARI
    a(n) = sum(k=1, n, gcd(k, n)^5);
  • PARI
    a(n) = sumdiv(n, d, eulerphi(n/d)*d^5);
  • PARI
    a(n) = sumdiv(n, d, moebius(n/d)*d*sigma(d, 4));
  • PARI
    my(N=40, x='x+O('x^N)); Vec(sum(k=1, N, eulerphi(k)*x^k*(1+26*x^k+66*x^(2*k)+26*x^(3*k)+x^(4*k))/(1-x^k)^6))
  • SageMath
    def A343499(n): return sum(k^5*euler_phi(n/k) for k in (1..n) if (k).divides(n))
[A343499(n) for n in range(1,51)] # G. C. Greubel, Jun 24 2024

Formula

a(n) = Sum_{d|n} phi(n/d) * d^5.
a(n) = Sum_{d|n} mu(n/d) * d * sigma_4(d).
G.f.: Sum_{k >= 1} phi(k) * x^k * (1 + 26*x^k + 66*x^(2*k) + 26*x^(3*k) + x^(4*k))/(1 - x^k)^6.
Dirichlet g.f.: zeta(s-1) * zeta(s-5) / zeta(s). - Ilya Gutkovskiy, Apr 18 2021
Sum_{k=1..n} a(k) ~ 315*zeta(5)*n^6 / (2*Pi^6). - Vaclav Kotesovec, May 20 2021
Multiplicative with a(p^e) = p^(e-1)*(p^(4*e+5) - p^(4*e) - p + 1)/(p^4-1). - Amiram Eldar, Nov 22 2022
a(n) = Sum_{1 <= i_1, ..., i_5 <= n} gcd(i_1, ..., i_5, n) = Sum_{d divides n} d * J_5(n/d), where the Jordan totient function J_5(n) = A059378(n). - Peter Bala, Jan 29 2024

A344479 Square array T(n,k), n >= 1, k >= 1, read by antidiagonals, where T(n,k) = Sum_{1 <= x_1, x_2, ..., x_k <= n} gcd(x_1, x_2, ..., x_k).

Original entry on oeis.org

1, 1, 3, 1, 5, 6, 1, 9, 12, 10, 1, 17, 30, 24, 15, 1, 33, 84, 76, 37, 21, 1, 65, 246, 276, 141, 61, 28, 1, 129, 732, 1060, 649, 267, 80, 36, 1, 257, 2190, 4164, 3165, 1417, 400, 112, 45, 1, 513, 6564, 16516, 15697, 8091, 2528, 624, 145, 55, 1, 1025, 19686, 65796, 78261, 47521, 17128, 4432, 885, 189, 66
Offset: 1

Views

Author

Seiichi Manyama, May 22 2021

Keywords

Examples

			G.f. of column 3: (1/(1 - x)) * Sum_{i>=1} phi(i) * (x^i + 4*x^(2*i) + x^(3*i))/(1 - x^i)^3.
Square array begins:
   1,  1,   1,    1,    1,     1, ...
   3,  5,   9,   17,   33,    65, ...
   6, 12,  30,   84,  246,   732, ...
  10, 24,  76,  276, 1060,  4164, ...
  15, 37, 141,  649, 3165, 15697, ...
  21, 61, 267, 1417, 8091, 47521, ...

Links

Crossrefs

Columns k=1..5 give A000217, A018806, A344522, A344523, A344524.
T(n,n) gives A344525.
Cf. A343510, A343516, A344527.

Programs

  • Mathematica
    T[n_, k_] := Sum[EulerPhi[j] * Quotient[n, j]^k, {j, 1, n}]; Table[T[k, n - k + 1], {n, 1, 11}, {k, 1, n}] // Flatten (* Amiram Eldar, May 22 2021 *)
  • PARI
    T(n, k) = sum(j=1, n, eulerphi(j)*(n\j)^k);

Formula

G.f. of column k: (1/(1 - x)) * Sum_{i>=1} phi(i) * ( Sum_{j=1..k} A008292(k, j) * x^(i*j) )/(1 - x^i)^k.
T(n,k) = Sum_{j=1..n} phi(j) * floor(n/j)^k.

A344523 a(n) = Sum_{1 <= i, j, k, l <= n} gcd(i,j,k,l).

Original entry on oeis.org

1, 17, 84, 276, 649, 1417, 2528, 4432, 7033, 10905, 15556, 22836, 30673, 41729, 54944, 71968, 89969, 115457, 140820, 175444, 212537, 257113, 302720, 366160, 426505, 500873, 580676, 677108, 769761, 895377, 1008928, 1153120, 1300417, 1469073, 1640020, 1860340, 2054921
Offset: 1

Views

Author

Seiichi Manyama, May 22 2021

Keywords

Links

Crossrefs

Column k=4 of A344479.
Cf. A082540, A343498, A344138, A344522, A344524, A344525.

Programs

  • Mathematica
    a[n_] := Sum[EulerPhi[k] * Quotient[n, k]^4, {k, 1, n}]; Array[a, 50] (* Amiram Eldar, May 22 2021 *)
  • PARI
    a(n) = sum(i=1, n, sum(j=1, n, sum(k=1, n, sum(l=1, n, gcd([i, j, k, l])))));
  • PARI
    a(n) = sum(k=1, n, eulerphi(k)*(n\k)^4);
  • PARI
    my(N=40, x='x+O('x^N)); Vec(sum(k=1, N, eulerphi(k)*x^k*(1+11*x^k+11*x^(2*k)+x^(3*k))/(1-x^k)^4)/(1-x))

Formula

a(n) = Sum_{k=1..n} phi(k) * floor(n/k)^4.
G.f.: (1/(1 - x)) * Sum_{k >= 1} phi(k) * x^k * (1 + 11*x^k + 11*x^(2*k) + x^(3*k))/(1 - x^k)^4.
a(n) ~ 90 * zeta(3) * n^4 / Pi^4. - Vaclav Kotesovec, May 23 2021

A344525 a(n) = Sum_{1 <= x_1, x_2, ... , x_n <= n} gcd(x_1,x_2, ... ,x_n).

Original entry on oeis.org

1, 5, 30, 276, 3165, 47521, 826000, 16843792, 387723045, 10009889889, 285360865350, 8918311872516, 302888304741841, 11112685595264369, 437898699063881208, 18447025862624951488, 827242515246907227633, 39346558373191515582161
Offset: 1

Views

Author

Seiichi Manyama, May 22 2021

Keywords

Links

Crossrefs

Main diagonal of A344479.
Cf. A018806, A344140, A344522, A344523, A344524.

Programs

  • Mathematica
    a[n_] := Sum[EulerPhi[k] * Quotient[n, k]^n, {k, 1, n}]; Array[a, 20] (* Amiram Eldar, May 22 2021 *)
  • PARI
    a(n) = sum(k=1, n, eulerphi(k)*(n\k)^n);
  • Python
    from sympy import totient
def A344525(n): return sum(totient(k)*(n//k)**n for k in range(1,n+1)) # Chai Wah Wu, Aug 05 2024

Formula

a(n) = Sum_{k=1..n} phi(k) * floor(n/k)^n.
a(n) ~ n^n. - Vaclav Kotesovec, May 23 2021
Showing 1-5 of 5 results.

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