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.

A372931 a(n) = Sum_{1 <= x_1, x_2, x_3, x_4 <= n} gcd(x_1, x_2, x_3, x_4, n)^4.

Original entry on oeis.org

1, 31, 161, 736, 1249, 4991, 4801, 15616, 19521, 38719, 29281, 118496, 57121, 148831, 201089, 311296, 167041, 605151, 260641, 919264, 772961, 907711, 559681, 2514176, 1170625, 1770751, 2106081, 3533536, 1414561, 6233759, 1847041, 5963776, 4714241, 5178271, 5996449
Offset: 1

Views

Author

Seiichi Manyama, May 17 2024

Keywords

Links

Crossrefs

Column k=4 of A372938.
Cf. A343498, A372926, A372929, A372937.
Cf. A000005, A008683.
Cf. A001620, A013663.

Programs

  • Mathematica
    f[p_, e_] := (e - e/p^4 + 1) * p^(4*e); 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^4*numdiv(d));

Formula

a(n) = Sum_{d|n} mu(n/d) * d^4 * tau(d), where mu is the Moebius function A008683.
From Amiram Eldar, May 21 2024: (Start)
Multiplicative with a(p^e) = (e - e/p^4 + 1) * p^(4*e).
Dirichlet g.f.: zeta(s-4)^2/zeta(s).
Sum_{k=1..n} a(k) ~ (n^5/(5*zeta(5))) * (log(n) + 2*gamma - 1/5 - zeta'(5)/zeta(5)), where gamma is Euler's constant (A001620). (End)

A372926 a(n) = Sum_{1 <= x_1, x_2 <= n} gcd(x_1, x_2, n)^4.

Original entry on oeis.org

1, 19, 89, 316, 649, 1691, 2449, 5104, 7281, 12331, 14761, 28124, 28729, 46531, 57761, 81856, 83809, 138339, 130681, 205084, 217961, 280459, 280369, 454256, 406225, 545851, 590409, 773884, 708121, 1097459, 924481, 1310464, 1313729, 1592371, 1589401, 2300796
Offset: 1

Views

Author

Seiichi Manyama, May 17 2024

Keywords

Links

Crossrefs

Cf. A069097, A360428, A368743, A372927.
Cf. A343498, A372929, A372931.
Cf. A001157, A008683.
Cf. A002117, A013663.

Programs

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

Formula

a(n) = Sum_{1 <= x_1, x_2, x_3, x_4 <= n} gcd(x_1, x_2, x_3, x_4, n)^2.
a(n) = Sum_{d|n} mu(n/d) * d^2 * sigma_2(d), where mu is the Moebius function A008683.
From Amiram Eldar, May 21 2024: (Start)
Multiplicative with a(p^e) = p^(2*e-2) * (p^2 * (p^(2*e+2)-1) - p^(2*e) + 1)/(p^2-1).
Dirichlet g.f.: zeta(s-2)*zeta(s-4)/zeta(s).
Sum_{k=1..n} a(k) ~ c * n^5 / 5, where c = zeta(3)/zeta(5) = 1.1592484598... . (End)

A372928 a(n) = Sum_{1 <= x_1, x_2, x_3 <= n} gcd(x_1, x_2, x_3, n)^3.

Original entry on oeis.org

1, 15, 53, 176, 249, 795, 685, 1856, 2133, 3735, 2661, 9328, 4393, 10275, 13197, 18432, 9825, 31995, 13717, 43824, 36305, 39915, 24333, 98368, 46625, 65895, 76545, 120560, 48777, 197955, 59581, 176128, 141033, 147375, 170565, 375408, 101305, 205755, 232829, 462144
Offset: 1

Views

Author

Seiichi Manyama, May 17 2024

Keywords

Links

Crossrefs

Column k=3 of A372938.
Cf. A343497, A368743, A372929, A372930.
Cf. A000005, A008683.
Cf. A001620, A013662, A261506.

Programs

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

Formula

a(n) = Sum_{d|n} mu(n/d) * d^3 * tau(d), where mu is the Moebius function A008683.
From Amiram Eldar, May 21 2024: (Start)
Multiplicative with a(p^e) = (e - e/p^3 + 1) * p^(3*e).
Dirichlet g.f.: zeta(s-3)^2/zeta(s).
Sum_{k=1..n} a(k) ~ (n^4/(4*zeta(4))) * (log(n) + 2*gamma - 1/4 - zeta'(4)/zeta(4)), where gamma is Euler's constant (A001620). (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)

A372937 a(n) = Sum_{1 <= x_1, x_2, x_3, x_4 <= n} gcd(x_1, x_2, x_3, x_4, n)^5.

Original entry on oeis.org

1, 47, 323, 1744, 3749, 15181, 19207, 59648, 84969, 176203, 175691, 563312, 399853, 902729, 1210927, 1970176, 1503377, 3993543, 2606419, 6538256, 6203861, 8257477, 6716183, 19266304, 12105625, 18793091, 21172347, 33497008, 21218429, 56913569, 29552671, 64028672
Offset: 1

Views

Author

Seiichi Manyama, May 17 2024

Keywords

Links

Crossrefs

Cf. A343498, A372926, A372929, A372931.
Cf. A000203, A008683.
Cf. A013661, A013664, A157292.

Programs

  • Mathematica
    f[p_, e_] := p^(4*e-4)*(p^e*(p^5-1) - (p^4-1))/(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^4*sigma(d));

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)^4.
a(n) = Sum_{d|n} mu(n/d) * d^4 * sigma(d), where mu is the Moebius function A008683.
From Amiram Eldar, May 21 2024: (Start)
Multiplicative with a(p^e) = p^(4*e-4)*(p^e*(p^5-1) - (p^4-1))/(p-1).
Dirichlet g.f.: zeta(s-4)*zeta(s-5)/zeta(s).
Sum_{k=1..n} a(k) ~ c * n^6 / 6, c = zeta(2)/zeta(6) = 315/(2*Pi^4) = 1.616892... (A157292). (End)
Mobius transformation of A280022. - R. J. Mathar, Jul 14 2025
Showing 1-5 of 5 results.

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