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.

Previous Showing 11-13 of 13 results.

A343508 a(n) = Sum_{k=1..n} gcd(k, n)^6.

Original entry on oeis.org

1, 65, 731, 4162, 15629, 47515, 117655, 266372, 532905, 1015885, 1771571, 3042422, 4826821, 7647575, 11424799, 17047816, 24137585, 34638825, 47045899, 65047898, 86005805, 115152115, 148035911, 194717932, 244203145, 313743365, 388487763, 489680110, 594823349
Offset: 1

Views

Author

Seiichi Manyama, Apr 17 2021

Keywords

Links

Crossrefs

Column 6 of A343510.
Cf. A000010, A001160 (sigma_5(n)), A069091, A343520.

Programs

  • Mathematica
    a[n_] := Sum[GCD[k, n]^6, {k, 1, n}]; Array[a, 50] (* Amiram Eldar, Apr 18 2021 *)
f[p_, e_] := p^(e-1)*(p^(5*e+6) - p^(5*e) - p + 1)/(p^5-1); 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)^6);
  • PARI
    a(n) = sumdiv(n, d, eulerphi(n/d)*d^6);
  • PARI
    a(n) = sumdiv(n, d, moebius(n/d)*d*sigma(d, 5));
  • PARI
    my(N=40, x='x+O('x^N)); Vec(sum(k=1, N, eulerphi(k)*x^k*(1+57*x^k+302*x^(2*k)+302*x^(3*k)+57*x^(4*k)+x^(5*k))/(1-x^k)^7))

Formula

a(n) = Sum_{d|n} phi(n/d) * d^6.
a(n) = Sum_{d|n} mu(n/d) * d * sigma_5(d).
G.f.: Sum_{k >= 1} phi(k) * x^k * (1 + 57*x^k + 302*x^(2*k) + 302*x^(3*k) + 57*x^(4*k) + x^(5*k))/(1 - x^k)^7.
Dirichlet g.f.: zeta(s-1) * zeta(s-6) / zeta(s). - Ilya Gutkovskiy, Apr 18 2021
Sum_{k=1..n} a(k) ~ Pi^6 * n^7 / (6615*zeta(7)). - Vaclav Kotesovec, May 20 2021
Multiplicative with a(p^e) = p^(e-1)*(p^(5*e+6) - p^(5*e) - p + 1)/(p^5-1). - Amiram Eldar, Nov 22 2022
a(n) = Sum_{1 <= i_1, ..., i_6 <= n} gcd(i_1, ..., i_6, n) = Sum_{d divides n} d * J_6(n/d), where the Jordan totient function J_6(n) = A069091(n). - Peter Bala, Jan 29 2024

A343509 a(n) = Sum_{k=1..n} gcd(k, n)^7.

Original entry on oeis.org

1, 129, 2189, 16514, 78129, 282381, 823549, 2113796, 4787349, 10078641, 19487181, 36149146, 62748529, 106237821, 171024381, 270565896, 410338689, 617568021, 893871757, 1290222306, 1802748761, 2513846349, 3404825469, 4627099444, 6103828145, 8094560241
Offset: 1

Views

Author

Seiichi Manyama, Apr 17 2021

Keywords

Comments

In general, for m > 1, if a(n) = Sum_{j=1..n} gcd(j, n)^m, then Sum_{k=1..n} a(k) ~ zeta(m) * n^(m+1) / ((m+1) * zeta(m+1)). - Vaclav Kotesovec, May 20 2021

Links

Crossrefs

Column 7 of A343510.
Cf. A000010, A013954 (sigma_6(n)), A069092, A343521.

Programs

  • Mathematica
    a[n_] := Sum[GCD[k, n]^7, {k, 1, n}]; Array[a, 50] (* Amiram Eldar, Apr 18 2021 *)
f[p_, e_] := p^(e-1)*(p^(6*e+7) - p^(6*e) - p + 1)/(p^6-1); 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)^7);
  • PARI
    a(n) = sumdiv(n, d, eulerphi(n/d)*d^7);
  • PARI
    a(n) = sumdiv(n, d, moebius(n/d)*d*sigma(d, 6));
  • PARI
    my(N=40, x='x+O('x^N)); Vec(sum(k=1, N, eulerphi(k)*x^k*(1+120*x^k+1191*x^(2*k)+2416*x^(3*k)+1191*x^(4*k)+120*x^(5*k)+x^(6*k))/(1-x^k)^8))

Formula

a(n) = Sum_{d|n} phi(n/d) * d^7.
a(n) = Sum_{d|n} mu(n/d) * d * sigma_6(d).
G.f.: Sum_{k >= 1} phi(k) * x^k * (1 + 120*x^k + 1191*x^(2*k) + 2416*x^(3*k) + 1191*x^(4*k) + 120*x^(5*k) + x^(6*k))/(1 - x^k)^8.
Dirichlet g.f.: zeta(s-1) * zeta(s-7) / zeta(s). - Ilya Gutkovskiy, Apr 18 2021
Sum_{k=1..n} a(k) ~ 4725*zeta(7)*n^8 / (4*Pi^8). - Vaclav Kotesovec, May 20 2021
Multiplicative with a(p^e) = p^(e-1)*(p^(6*e+7) - p^(6*e) - p + 1)/(p^6-1). - Amiram Eldar, Nov 22 2022
a(n) = Sum_{1 <= i_1, ..., i_7 <= n} gcd(i_1, ..., i_7, n) = Sum_{d divides n} d * J_7(n/d), where the Jordan totient function J_7(n) = A069092(n). - Peter Bala, Jan 29 2024

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)
Previous Showing 11-13 of 13 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

Content is available under The OEIS End-User License Agreement.