Year-end appeal: Please make a donation to the OEIS Foundation to support ongoing development and maintenance of the OEIS. We are now in our 61st year, we have over 378,000 sequences, and we’ve reached 11,000 citations (which often say “discovered thanks to the OEIS”).
%I #36 Nov 04 2022 14:44:47
%S 1,5,9,19,23,39,43,63,73,89,93,133,137,153,169,204,208,248,252,292,
%T 308,324,328,408,418,434,454,494,498,562,566,622,638,654,670,770,774,
%U 790,806,886,890,954,958,998,1038,1054,1058,1198,1208,1248,1264,1304,1308
%N (tau<=)_4(n).
%C (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.
%C Partial sums of A007426.
%C Equals row sums of triangle A140703. - _Gary W. Adamson_, May 24 2008
%H Vaclav Kotesovec, <a href="/A061202/b061202.txt">Table of n, a(n) for n = 1..10000</a>
%H Vaclav Kotesovec, <a href="/A061202/a061202.jpg">Graph - The asymptotic ratio (1000000 terms)</a>
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/StieltjesConstants.html">Stieltjes Constants</a>
%H Wikipedia, <a href="http://en.wikipedia.org/wiki/Stieltjes_constants">Stieltjes constants</a>
%F (tau<=)_k(n) = Sum_{i=1..n} tau_k(i).
%F a(n) = Sum_{k = 1..n} tau_{3}(k)*floor (n/k), where tau_{3} is A007425. - _Enrique Pérez Herrero_, Jan 23 2013
%F 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
%F a(n) = Sum_{i=1..n} tau(i)*A006218(floor(n/i)). - _Ridouane Oudra_, Sep 17 2021
%F a(n) = Sum_{i=1..n} Sum_{j=1..n} Sum_{k=1..n} floor(n/(i*j*k)). - _Ridouane Oudra_, Oct 31 2022
%t (* 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 *)
%Y 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.
%Y Equals left column of triangle A140705.
%Y Cf. A140703.
%K easy,nonn
%O 1,2
%A _Vladeta Jovovic_, Apr 21 2001