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 1-4 of 4 results.

A372929 a(n) = Sum_{1 <= x_1, x_2, x_3 <= n} gcd(x_1, x_2, x_3, n)^4.

Original entry on oeis.org

1, 23, 107, 424, 749, 2461, 2743, 7232, 9369, 17227, 15971, 45368, 30757, 63089, 80143, 119296, 88433, 215487, 137179, 317576, 293501, 367333, 292007, 773824, 483625, 707411, 777843, 1163032, 731669, 1843289, 953311, 1937408, 1708897, 2033959, 2054507, 3972456
Offset: 1

Views

Author

Seiichi Manyama, May 17 2024

Keywords

Links

Crossrefs

Cf. A343497, A368743, A372928, A372930.
Cf. A343498, A372926, A372931.
Cf. A000203, A008683.
Cf. A013661, A013663.

Programs

  • Mathematica
    f[p_, e_] := p^(3*e-3) * (p^3 * (p^(e+1)-1) - p^e + 1)/(p-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)*d^3*sigma(d));

Formula

a(n) = Sum_{1 <= x_1, x_2, x_3, x_4 <= n} gcd(x_1, x_2, x_3, x_4, n)^3.
a(n) = Sum_{d|n} mu(n/d) * d^3 * sigma(d), where mu is the Moebius function A008683.
From Amiram Eldar, May 21 2024: (Start)
Multiplicative with a(p^e) = p^(3*e-3) * (p^3 * (p^(e+1)-1) - p^e + 1)/(p-1).
Dirichlet g.f.: zeta(s-3)*zeta(s-4)/zeta(s).
Sum_{k=1..n} a(k) ~ c * n^5 / 5, where c = zeta(2)/zeta(5) = 1.586353589... . (End)

A372930 a(n) = Sum_{1 <= x_1, x_2, x_3 <= n} gcd(x_1, x_2, x_3, n)^5.

Original entry on oeis.org

1, 39, 269, 1304, 3249, 10491, 17149, 42176, 66069, 126711, 162381, 350776, 373489, 668811, 873981, 1353216, 1424769, 2576691, 2482957, 4236696, 4613081, 6332859, 6448509, 11345344, 10168625, 14566071, 16073721, 22362296, 20535537, 34085259, 28658941, 43331584
Offset: 1

Views

Author

Seiichi Manyama, May 17 2024

Keywords

Links

Crossrefs

Cf. A343497, A368743, A372928, A372929.
Cf. A001157, A008683.
Cf. A002117, A013664, A157289.

Programs

  • Mathematica
    f[p_, e_] := p^(3*e-3) * (p^3 * (p^(2*e+2)-1) - p^(2*e) + 1)/(p^2-1); 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)*d^3*sigma(d, 2));

Formula

a(n) = Sum_{1 <= x_1, x_2, x_3, x_4, x_5 <= n} gcd(x_1, x_2, x_3, x_4, x_5, n)^3.
a(n) = Sum_{d|n} mu(n/d) * d^3 * sigma_2(d), where mu is the Moebius function A008683.
From Amiram Eldar, May 21 2024: (Start)
Multiplicative with a(p^e) = p^(3*e-3) * (p^3 * (p^(2*e+2)-1) - p^(2*e) + 1)/(p^2-1).
Dirichlet g.f.: zeta(s-3)*zeta(s-5)/zeta(s).
Sum_{k=1..n} a(k) ~ c * n^6 / 6, where c = zeta(3)/zeta(6) = 1.181564... (A157289). (End)

A372938 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, n)^k.

Original entry on oeis.org

1, 1, 3, 1, 7, 5, 1, 15, 17, 8, 1, 31, 53, 40, 9, 1, 63, 161, 176, 49, 15, 1, 127, 485, 736, 249, 119, 13, 1, 255, 1457, 3008, 1249, 795, 97, 20, 1, 511, 4373, 12160, 6249, 4991, 685, 208, 21, 1, 1023, 13121, 48896, 31249, 30555, 4801, 1856, 225, 27
Offset: 1

Views

Author

Seiichi Manyama, May 17 2024

Keywords

Examples

			Square array begins:
   1,   1,   1,    1,     1,      1,       1, ...
   3,   7,  15,   31,    63,    127,     255, ...
   5,  17,  53,  161,   485,   1457,    4373, ...
   8,  40, 176,  736,  3008,  12160,   48896, ...
   9,  49, 249, 1249,  6249,  31249,  156249, ...
  15, 119, 795, 4991, 30555, 185039, 1115115, ...
  13,  97, 685, 4801, 33613, 235297, 1647085, ...

Crossrefs

Columns k=1..4 give: A018804, A360428, A372928, A372931.
Main diagonal gives A372939.
Cf. A000005, A001620, A008683, A343510.

Programs

  • Mathematica
    f[p_, e_, k_] := (e - e/p^k + 1)*p^(k*e); T[1, k_] := 1; T[n_, k_] := Times @@ (f[First[#], Last[#], k] & /@ FactorInteger[n]); Table[T[k, n - k + 1], {n, 1, 10}, {k, 1, n}] // Flatten (* Amiram Eldar, May 25 2024 *)
  • PARI
    T(n,k) = sumdiv(n, d, moebius(n/d)*d^k*numdiv(d));

Formula

a(n) = Sum_{d|n} mu(n/d) * d^k * tau(d), where mu is the Moebius function A008683.
From Amiram Eldar, May 25 2024: (Start)
T(n,k) for a given k is multiplicative with T(p^e, k) = (e - e/p^k + 1) * p^(k*e).
Dirichlet g.f. of T(n, k) for a given k: zeta(s-k)^2/zeta(s).
Sum_{m=1..n} T(m, k) ~ (n^(k+1)/((k+1)*zeta(k+1))) * (log(n) + 2*gamma - 1/(k+1) - zeta'(k+1)/zeta(k+1)), where gamma is Euler's constant (A001620). (End)

A386012 a(n) = n^3*tau(n).

Original entry on oeis.org

1, 16, 54, 192, 250, 864, 686, 2048, 2187, 4000, 2662, 10368, 4394, 10976, 13500, 20480, 9826, 34992, 13718, 48000, 37044, 42592, 24334, 110592, 46875, 70304, 78732, 131712, 48778, 216000, 59582, 196608, 143748, 157216, 171500, 419904, 101306, 219488, 237276, 512000
Offset: 1

Views

Author

R. J. Mathar, Jul 14 2025

Keywords

Comments

Dirichlet convolution of the cubes A000578 with themselves.

Links

Crossrefs

Cf. A000005, A001620, A034714, A038040, A320895 (partial sums), A372928 (Mobius transform).

Programs

  • Maple
    seq( n^3*numtheory[tau](n),n=1..100) ;
  • Mathematica
    a[n_]:=n^3*DivisorSigma[0,n]; Array[a,40] (* Stefano Spezia, Jul 14 2025 *)
nmax = 40; Rest[CoefficientList[Series[Sum[k^3*x^k*(1 + 4*x^k + x^(2*k))/(1 - x^k)^4, {k, 1, nmax}], {x, 0, nmax}], x]] (* Vaclav Kotesovec, Aug 03 2025 *)
  • PARI
    a(n) = n^3 * numdiv(n); \\ Amiram Eldar, Jul 15 2025

Formula

a(n) = n*A034714(n) = n^2*A038040(n).
Dirichlet g.f.: zeta^2(s-3).
From Amiram Eldar, Jul 15 2025 (Start)
Multiplicative with a(p^e) = p^(3*e) * (e+1).
Sum_{k=1..n} a(k) ~ (n^4/4) * (log(n) + 2*gamma - 1/4), where gamma is Euler's constant (A001620). (End)
G.f.: Sum_{k>=1} k^3*x^k*(1 + 4*x^k + x^(2*k)) / (1-x^k)^4. - Vaclav Kotesovec, Aug 03 2025
Showing 1-4 of 4 results.

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