A360428 -id:A360428 - cp's OEIS frontend

A069097 Moebius transform of A064987, n*sigma(n).

1, 5, 11, 22, 29, 55, 55, 92, 105, 145, 131, 242, 181, 275, 319, 376, 305, 525, 379, 638, 605, 655, 551, 1012, 745, 905, 963, 1210, 869, 1595, 991, 1520, 1441, 1525, 1595, 2310, 1405, 1895, 1991, 2668, 1721, 3025, 1891, 2882, 3045, 2755, 2255, 4136, 2737

Offset: 1

Benoit Cloitre, Apr 05 2002

Equals A127569 * [1, 2, 3, ...]. - Gary W. Adamson, Jan 19 2007
Equals row sums of triangle A143309 and of triangle A143312. - Gary W. Adamson, Aug 06 2008
Dirichlet convolution of A000290 and A000010 (see Jovovic formula). - R. J. Mathar, Feb 03 2011

Amiram Eldar, Table of n, a(n) for n = 1..10000
Wenchang Chu, Jordan's totient function and trigonometric sums, Studia Scientiarum Mathematicarum Hungarica 57 (1), 40-53 (2020).
Nikolai Osipov, Problem 12003, Amer. Math. Monthly, 124(8) (2017), p. 754.

Column 2 of A343510.
Cf. A018804, A033457, A068963, A064987, A127569, A143309, A143312, A360428.
For Sum_{k = 1..n} gcd(k,n)^m see A018804 (m = 1), A343497 (m = 3), A343498 (m = 4) and A343499 (m = 5).

A069097[n_]:=n^2*Plus @@((EulerPhi[#]/#^2)&/@ Divisors[n]); Array[A069097, 100] (* Enrique Pérez Herrero, Feb 25 2012 *)
f[p_, e_] := p^(e-1)*(p^e*(p+1)-1); a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Sep 18 2020 *)

for(n=1,100,print1((sumdiv(n,k,k*sigma(k)*moebius(n/k))),","))

a(n) = Sum_{d|n} d^2*phi(n/d). - Vladeta Jovovic, Jul 31 2002
a(n) = Sum_{k=1..n} gcd(n, k)^2. - Vladeta Jovovic, Aug 27 2003
Dirichlet g.f.: zeta(s-2)*zeta(s-1)/zeta(s). - R. J. Mathar, Feb 03 2011
a(n) = n*Sum_{d|n} J_2(d)/d, where J_2 is A007434. - Enrique Pérez Herrero, Feb 25 2012.
G.f.: Sum_{n >= 1} phi(n)*(x^n + x^(2*n))/(1 - x^n)^3 = x + 5*x^2 + 11*x^3 + 22*x^4  + .... - Peter Bala, Dec 30 2013
Multiplicative with a(p^e) = p^(e-1)*(p^e*(p+1)-1). - R. J. Mathar, Jun 23 2018
Sum_{k=1..n} a(k) ~ Pi^2 * n^3 / (18*zeta(3)). - Vaclav Kotesovec, Sep 18 2020
a(n) = Sum_{k=1..n} (n/gcd(n,k))^2*phi(gcd(n,k))/phi(n/gcd(n,k)). - Richard L. Ollerton, May 07 2021
From Peter Bala, Dec 26 2023: (Start)
For n odd, a(n) = Sum_{k = 1..n} gcd(k,n)/cos(k*Pi/n)^2 (see Osipov and also Chu, p. 51).
It appears that for n odd, Sum_{k = 1..n} (-1)^(k+1)*gcd(k,n)/cos(k*Pi/n)^2 = n. (End)
a(n) = Sum_{1 <= i, j <= n} gcd(i, j, n). Cf. A360428. - Peter Bala, Jan 16 2024
Sum_{k=1..n} a(k)/k ~ Pi^2 * n^2 / (12*zeta(3)). - Vaclav Kotesovec, May 11 2024

A368743 a(n) = Sum_{1 <= i, j <= n} gcd(i, j, n)^3.

Original entry on oeis.org

1, 11, 35, 100, 149, 385, 391, 848, 1017, 1639, 1451, 3500, 2365, 4301, 5215, 6976, 5201, 11187, 7219, 14900, 13685, 15961, 12695, 29680, 19225, 26015, 28107, 39100, 25229, 57365, 30751, 56576, 50785, 57211, 58259, 101700, 52021, 79409, 82775, 126352

Offset: 1

Peter Bala, Jan 20 2024

Amiram Eldar, Table of n, a(n) for n = 1..10000
Peter Bala, GCD sum theorems. Two Multivariable Cesaro Type Identities.

Cf. A007434, A059376, A069097, A343497, A343498, A360428.

seq( add(add(igcd(i, j, n)^3, i = 1..n), j = 1..n), n = 1..50);
# faster program for large n
with(numtheory):
A007434 := proc(n) add(d^2*mobius(n/d), d in divisors(n)) end proc:
seq( add(d^3*A007434(n/d), d in divisors(n)), n = 1..500);

f[p_, e_] := p^(3*e - 2)*(p^2 + p + 1) - p^(2*e - 2)*(p + 1); a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Jan 29 2024 *)

a(n) = {my(f = factor(n)); prod(i = 1, #f~, p = f[i,1]; e = f[i,2]; p^(3*e - 2)*(p^2 + p + 1) - p^(2*e - 2)*(p + 1));} \\ Amiram Eldar, Jan 29 2024

from math import prod
from sympy import factorint
def A368743(n): return prod(p**(e-1<<1)*(p**e*(p*(q:=p+1)+1)-q) for p, e in factorint(n).items()) # Chai Wah Wu, Jan 29 2024

a(n) = Sum_{1 <= i, j, k <= n} gcd(i, j, k, n)^2.
a(n) = Sum_{d divides n} d^3 * J_2(n/d) = Sum_{d divides n} d^2 * J_3(n/d), where the Jordan totient functions J_2(n) = A007434(n) and J_3(n) = A059376(n).
Dirichlet g.f.: zeta(s-2) * zeta(s-3)/zeta(s).
a(n) is a multiplicative function: for prime p, a(p^k) = p^(3*k-2)*(p^2 + p + 1) - p^(2*k-2)*(p + 1).
Sum_{k=1..n} a(k) ~ c * n^4, where c = 15/(4*Pi^2) = 0.379954... . - Amiram Eldar, Jan 29 2024
a(n) = Sum_{d divides n} mobius(n/d) * d^2 * sigma(d). - Peter Bala, Jan 29 2024

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

Seiichi Manyama, May 17 2024

Seiichi Manyama, Table of n, a(n) for n = 1..10000
Peter Bala, GCD sum theorems. Two Multivariable Cesaro Type Identities.

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

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 *)

a(n) = sumdiv(n, d, moebius(n/d)*d^2*sigma(d, 2));

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)

A368929 Dirichlet g.f.: zeta(s-2)^2 * (1 - 2^(3-s)) / zeta(s).

Original entry on oeis.org

1, -1, 17, -16, 49, -17, 97, -112, 225, -49, 241, -272, 337, -97, 833, -640, 577, -225, 721, -784, 1649, -241, 1057, -1904, 1825, -337, 2673, -1552, 1681, -833, 1921, -3328, 4097, -577, 4753, -3600, 2737, -721, 5729, -5488, 3361, -1649, 3697, -3856, 11025, -1057, 4417

Offset: 1

Vaclav Kotesovec, Jan 12 2024

Dirichlet convolution of A007434 and A162395.

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A048272, A332794.

Cf. A007434, A162395, A360428.

Table[Sum[Sum[d^2 * MoebiusMu[k/d], {d, Divisors[k]}] * (-1)^(n/k + 1) * n^2/k^2, {k, Divisors[n]}], {n, 1, 100}]
f[p_, e_] := p^(2*e)*(1 + e*(1 - 1/p^2)); f[2, e_] := -(3*e - 2)*2^(2*e - 2); a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Jan 12 2024 *)

a(n) = {my(f = factor(n), p, e); prod(i = 1, #f~, p = f[i, 1]; e = f[i, 2]; if(p == 2, -(3*e-2)*2^(2*e-2), p^(2*e)*(1 + e*(1-1/p^2))));} \\ Amiram Eldar, Jan 12 2024

Sum_{k=1..n} a(k) ~ log(2) * n^3 / (3*zeta(3)).

Multiplicative with a(2^e) = -(3*e-2)*2^(2*e-2), and a(p^e) = p^(2*e)*(1 + e*(1-1/p^2)) for an odd prime p. - Amiram Eldar, Jan 12 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

Seiichi Manyama, May 17 2024

			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, ...

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

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 *)

T(n,k) = sumdiv(n, d, moebius(n/d)*d^k*numdiv(d));

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)

A372927 a(n) = Sum_{1 <= x_1, x_2 <= n} gcd(x_1, x_2, n)^5.

Original entry on oeis.org

1, 35, 251, 1132, 3149, 8785, 16855, 36272, 61065, 110215, 161171, 284132, 371461, 589925, 790399, 1160896, 1420145, 2137275, 2476459, 3564668, 4230605, 5640985, 6436871, 9104272, 9841225, 13001135, 14839443, 19079860, 20511989, 27663965, 28630111, 37149440, 40453921

Offset: 1

Seiichi Manyama, May 17 2024

Seiichi Manyama, Table of n, a(n) for n = 1..10000
Peter Bala, GCD sum theorems. Two Multivariable Cesaro Type Identities.

Cf. A069097, A360428, A368743, A372926.
Cf. A001158, A008683.
Cf. A013662, A013664, A088246.

f[p_, e_] := p^(2*e-2) * (p^2 * (p^(3*e+3)-1) - p^(3*e) + 1)/(p^3-1); a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, May 21 2024 *)

a(n) = sumdiv(n, d, moebius(n/d)*d^2*sigma(d, 3));

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)^2.
a(n) = Sum_{d|n} mu(n/d) * d^2 * sigma_3(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^(3*e+3)-1) - p^(3*e) + 1)/(p^3-1).
Dirichlet g.f.: zeta(s-2)*zeta(s-5)/zeta(s).
Sum_{k=1..n} a(k) ~ c * n^6 / 6, where c = zeta(4)/zeta(6) = 21/(2*Pi^2) = 1.0638724... (A088246). (End)

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A069097 Moebius transform of A064987, n*sigma(n).

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A368743 a(n) = Sum_{1 <= i, j <= n} gcd(i, j, n)^3.

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

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A368929 Dirichlet g.f.: zeta(s-2)^2 * (1 - 2^(3-s)) / zeta(s).

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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.

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A372927 a(n) = Sum_{1 <= x_1, x_2 <= n} gcd(x_1, x_2, n)^5.

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