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.
%I A189020 #18 Feb 08 2025 09:00:37
%S A189020 1,89,3575,93237,1951526,35270969,578262093,8840109380,128217432396,
%T A189020 1784942188189,24045237260214,315312623543840,4042957241191810,
%U A189020 50862246063060180,629513636928477232,7681900592647818929,92587253467765253144,1103781870246459696784,13031388731053572679450,152516435040764735691556,1771079109308495896176156
%N A189020 a(n) = Sum_{k=1..10^n} tau_4(k), where tau_4 is the number of ordered factorizations into 4 factors (A007426).
%C A189020 Using that tau_4 = tau_2 ** tau_2, where ** means Dirichlet convolution and tau_2 is (A000005), one can calculate a(n) faster than in O(10^n) operations - namely in O(10^(3n/4)) or even in O(10^(2n/3)). See links for details.
%H A189020 A. V. Lelechenko, <a href="http://taac.org.ua/files/a2011/proceedings/UA-1-Andrew%20Vladimirovich%20Lelechenko-83.pdf">The summation of the multiplicative functions</a> (in Russian).
%F A189020 a(n) = A061202(10^n) = Sum_{k=1..10^n} A007426(n).
%Y A189020 Cf. A057494 - partial sums up to 10^n of the divisors function tau_2 (A000005), A180361 - of the unitary divisors function tau_2* (A034444), A180365 - of the 3-divisors function tau_3 (A007425).
%Y A189020 Also see A072692 for such sums of the sum of divisors function (A000203), A084237 for sums of Moebius function (A008683), A064018 for sums of Euler totient function (A000010).
%K A189020 nonn
%O A189020 0,2
%A A189020 _Andrew Lelechenko_, Apr 15 2011
%E A189020 a(16)-a(20) from _Henri Lifchitz_, Feb 05 2025