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A061203 (tau<=)_5(n).

%I A061203 #26 Feb 13 2022 23:15:20
%S A061203 1,6,11,26,31,56,61,96,111,136,141,216,221,246,271,341,346,421,426,
%T A061203 501,526,551,556,731,746,771,806,881,886,1011,1016,1142,1167,1192,
%U A061203 1217,1442,1447,1472,1497,1672,1677,1802,1807,1882,1957,1982,1987,2337,2352
%N A061203 (tau<=)_5(n).
%C A061203 (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 A061203 Partial sums of A061200.
%C A061203 Equals row sums of triangle A140705. - _Gary W. Adamson_, May 24 2008
%H A061203 Vaclav Kotesovec, <a href="/A061203/b061203.txt">Table of n, a(n) for n = 1..10000</a>
%H A061203 Vaclav Kotesovec, <a href="/A061203/a061203.jpg">Graph - The asymptotic ratio (100000 terms)</a>
%F A061203 (tau<=)_k(n) = Sum_{i=1..n} tau_k(i).
%F A061203 a(n) = Sum_{k=1..n} tau_{4}(k) * floor(n/k), where tau_{4} is A007426. - _Enrique Pérez Herrero_, Jan 23 2013
%F A061203 a(n) ~ n*(log(n)^4/24 + (5*g/6 - 1/6)*log(n)^3 + 10*g1^2 + (5*g^2 - 5*g/2 - 5*g1/2 + 1/2)*log(n)^2 + (10*g^3 - 10*g^2 + (5 - 20*g1)*g + 5*g1 + 5*g2/2 - 1)*log(n) + 5*g^4 - 10*g^3 + (10 - 30*g1)*g^2 + (20*g1 + 10*g2 - 5)*g - 5*g1 - 5*g2/2 - 5*g3/6 + 1), where g is the Euler-Mascheroni constant A001620 and g1, g2, g3 are the Stieltjes constants, see A082633, A086279 and A086280. - _Vaclav Kotesovec_, Sep 10 2018
%p A061203 b:= proc(k, n) option remember; uses numtheory;
%p A061203      `if`(k=1, 1, add(b(k-1, d), d=divisors(n)))
%p A061203     end:
%p A061203 a:= proc(n) option remember; `if`(n=0, 0, b(5, n)+a(n-1)) end:
%p A061203 seq(a(n), n=1..49);  # _Alois P. Heinz_, Feb 13 2022
%t A061203 nmax = 50;
%t A061203 tau4 = Table[DivisorSum[n, DivisorSigma[0, n/#]*DivisorSigma[0, #] &], {n, 1, nmax}];
%t A061203 Accumulate[Table[Sum[tau4[[d]], {d, Divisors[n]}], {n, nmax}]] (* _Vaclav Kotesovec_, Sep 10 2018 *)
%Y A061203 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<=)_4(n): A061202, (tau<=)_6(n): A061204.
%Y A061203 Cf. A140705.
%K A061203 nonn,easy
%O A061203 1,2
%A A061203 _Vladeta Jovovic_, Apr 21 2001