A061202 (tau<=)_4(n).

1, 5, 9, 19, 23, 39, 43, 63, 73, 89, 93, 133, 137, 153, 169, 204, 208, 248, 252, 292, 308, 324, 328, 408, 418, 434, 454, 494, 498, 562, 566, 622, 638, 654, 670, 770, 774, 790, 806, 886, 890, 954, 958, 998, 1038, 1054, 1058, 1198, 1208, 1248, 1264, 1304, 1308

Offset

1,2

Comments

(tau<=)_k(n) = |{(x_1,x_2,...,x_k): x_1*x_2*...*x_k <= n}|, i.e., (tau<=)_k(n) is number of solutions to x_1*x_2*...*x_k <= n, x_i > 0.

Partial sums of A007426.

Equals row sums of triangle A140703. - Gary W. Adamson, May 24 2008

Links

Vaclav Kotesovec, Table of n, a(n) for n = 1..10000

Vaclav Kotesovec, Graph - The asymptotic ratio (1000000 terms)

Eric Weisstein's World of Mathematics, Stieltjes Constants

Wikipedia, Stieltjes constants

Formula

(tau<=)_k(n) = Sum_{i=1..n} tau_k(i).

a(n) = Sum_{k = 1..n} tau_{3}(k)*floor (n/k), where tau_{3} is A007425. - Enrique Pérez Herrero, Jan 23 2013

a(n) ~ n * (log(n)^3/6 + (2*g - 1/2)*log(n)^2 + (6*g^2 - 4*g - 4*g1 + 1)*log(n) + 4*g^3 - 6*g^2 + 4*g + 4*g1*(1 - 3*g) + 2*g2 - 1), where g is the Euler-Mascheroni constant A001620, g1 and g2 are the Stieltjes constants, see A082633 and A086279. - Vaclav Kotesovec, Sep 09 2018

a(n) = Sum_{i=1..n} tau(i)*A006218(floor(n/i)). - Ridouane Oudra, Sep 17 2021

a(n) = Sum_{i=1..n} Sum_{j=1..n} Sum_{k=1..n} floor(n/(i*j*k)). - Ridouane Oudra, Oct 31 2022

Mathematica

(* Asymptotics: *) n * (Log[n]^3/6 + (2*EulerGamma - 1/2)*Log[n]^2 + (6*EulerGamma^2 - 4*EulerGamma - 4*StieltjesGamma[1] + 1)*Log[n] + 4*EulerGamma^3 - 6*EulerGamma^2 + 4*EulerGamma + 4*StieltjesGamma[1]*(1 - 3*EulerGamma) + 2*StieltjesGamma[2] - 1) (* Vaclav Kotesovec, Sep 09 2018 *)

Crossrefs

Cf. tau_2(n): A000005, tau_3(n): A007425, tau_4(n): A007426, tau_5(n): A061200, tau_6(n): A034695, (unordered) 2-factorizations of n: A038548, (unordered) 3-factorizations of n: A034836, A001055, (tau<=)_2(n): A006218, (tau<=)_3(n): A061201, (tau<=)_5(n): A061203, (tau<=)_6(n): A061204.

Equals left column of triangle A140705.

Cf. A140703.

Sequence in context: A113805 A160722 A255652 * A235799 A295966 A060161

Adjacent sequences: A061199 A061200 A061201 * A061203 A061204 A061205

Keyword

easy,nonn

Author

Vladeta Jovovic, Apr 21 2001

Status

approved