%I #26 Mar 12 2023 07:34:23
%S 1,9,17,44,52,116,124,188,215,279,287,503,511,575,639,764,772,988,996,
%T 1212,1276,1340,1348,1860,1887,1951,2015,2231,2239,2751,2759,2975,
%U 3039,3103,3167,3896,3904,3968,4032,4544,4552,5064,5072,5288,5504,5568,5576
%N a(n) = Sum_{k=1..n} tau(k)^3, where tau is A000005.
%H Vaclav Kotesovec, <a href="/A318755/b318755.txt">Table of n, a(n) for n = 1..10000</a>
%H Vaclav Kotesovec, <a href="/A318755/a318755.jpg">Graph - the asymptotic ratio (100000 terms)</a>
%H Ramanujan's Papers, <a href="http://ramanujan.sirinudi.org/Volumes/published/ram17.html">Some formulas in the analytic theory of numbers</a>, Messenger of Mathematics, XLV, 1916, 81-84.
%F a(n) ~ n * (A1*log(n)^7 + A2*log(n)^6 + A3*log(n)^5 + A4*log(n)^4 + A5*log(n)^3 + A6*log(n)^2 + A7*log(n) + A8) [Ramanujan, 1916, formula (8)].
%F From _Vaclav Kotesovec_, Mar 12 2023: (Start)
%F Let f(s) = Product_{p prime} (1 - 9/p^(2*s) + 16/p^(3*s) - 9/p^(4*s) + 1/p^(6*s)), then
%F A1 = f(1)/5040 = 0.0000097860463451190658257888710490039661018239924009134296302566263529129...
%F A2 = ((8*gamma - 1)*f(1) + f'(1)) / 720 = 0.0007019997226174095261771358653540021199703406583347258622085873074052900...
%F A3 = (2 * f(1) * (1 - 8*gamma + 28*gamma^2 - 8*sg1) + 2*(8*gamma - 1)*f'(1) + f''(1)) / 240 = 0.0171707557268638504150726777646428533953516776541779590118582753709080243...
%F A4 = (6*f(1)*(-1 - 28*gamma^2 + 56*gamma^3 + gamma*(8 - 56*sg1) + 8*sg1 + 4*sg2) + 6*(1 - 8*gamma + 28*gamma^2 - 8*sg1)*f'(1) + (24*gamma - 3)*f''(1) + f'''(1)) / 144 = 0.1758477246705824231478998937203303065702508974398264386862202155788...,
%F where f(1) = Product_{p prime} (1 - 9/p^2 + 16/p^3 - 9/p^4 + 1/p^6) = 0.0493216735794000917619759100869799891531929217006036853364933968186814900...,
%F f'(1) = f(1) * Sum_{p prime} 6*(3*p + 1) * log(p) / ((p-1) * (p^2 + 4*p + 1)) = 0.3270075329904166293296173488834535949530448497141635531152019426434776932...,
%F f''(1) = f'(1)^2 / f(1) + f(1) * Sum_{p prime} (-36 * p^2 * (p+1)^2 * log(p)^2 / ((p-1)^2 * (p^2 + 4*p + 1)^2))) = 1.1340946589859924227356699847227569935993284591079455746283572890834872890...,
%F f'''(1) = 3*f'(1)*f''(1)/f(1) - 2*f'(1)^3/f(1)^2 + f(1) * Sum_{p prime} 72*p^2 * (p^5 + 3*p^4 + 8*p^3 + 8*p^2 + 3*p+ 1) * log(p)^3 / ((p-1)^3 * (p^2+ 4*p + 1)^3) = -1.3447542210274297874241826540796632006263184659735145444999327537246287...,
%F gamma is the Euler-Mascheroni constant A001620 and sg1, sg2 are the Stieltjes constants, see A082633 and A086279.
%F Approximate values of other constants:
%F A5 = 0.7626157870664479996781152281270580148665443022014605423466363134512...
%F A6 = 1.3720912878905940866975369743071441424192833481004753922122458993040...
%F A7 = 1.1416118168318711437057727816148048057614284471759625288073915723140...
%F A8 = 0.2618221765943171424958051160111945242076019991649774700610674747694...
%F (End)
%t Accumulate[DivisorSigma[0, Range[50]]^3]
%o (PARI) a(n) = sum(k=1, n, numdiv(k)^3); \\ _Michel Marcus_, Sep 03 2018
%Y Cf. A000005, A006218, A061502, A319089.
%Y Cf. A143127, A319085, A320895, A320896, A320897.
%K nonn
%O 1,2
%A _Vaclav Kotesovec_, Sep 02 2018