login

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”).

A318755
a(n) = Sum_{k=1..n} tau(k)^3, where tau is A000005.
5
1, 9, 17, 44, 52, 116, 124, 188, 215, 279, 287, 503, 511, 575, 639, 764, 772, 988, 996, 1212, 1276, 1340, 1348, 1860, 1887, 1951, 2015, 2231, 2239, 2751, 2759, 2975, 3039, 3103, 3167, 3896, 3904, 3968, 4032, 4544, 4552, 5064, 5072, 5288, 5504, 5568, 5576
OFFSET
1,2
LINKS
Ramanujan's Papers, Some formulas in the analytic theory of numbers, Messenger of Mathematics, XLV, 1916, 81-84.
FORMULA
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)].
From Vaclav Kotesovec, Mar 12 2023: (Start)
Let f(s) = Product_{p prime} (1 - 9/p^(2*s) + 16/p^(3*s) - 9/p^(4*s) + 1/p^(6*s)), then
A1 = f(1)/5040 = 0.0000097860463451190658257888710490039661018239924009134296302566263529129...
A2 = ((8*gamma - 1)*f(1) + f'(1)) / 720 = 0.0007019997226174095261771358653540021199703406583347258622085873074052900...
A3 = (2 * f(1) * (1 - 8*gamma + 28*gamma^2 - 8*sg1) + 2*(8*gamma - 1)*f'(1) + f''(1)) / 240 = 0.0171707557268638504150726777646428533953516776541779590118582753709080243...
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...,
where f(1) = Product_{p prime} (1 - 9/p^2 + 16/p^3 - 9/p^4 + 1/p^6) = 0.0493216735794000917619759100869799891531929217006036853364933968186814900...,
f'(1) = f(1) * Sum_{p prime} 6*(3*p + 1) * log(p) / ((p-1) * (p^2 + 4*p + 1)) = 0.3270075329904166293296173488834535949530448497141635531152019426434776932...,
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'''(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...,
gamma is the Euler-Mascheroni constant A001620 and sg1, sg2 are the Stieltjes constants, see A082633 and A086279.
Approximate values of other constants:
A5 = 0.7626157870664479996781152281270580148665443022014605423466363134512...
A6 = 1.3720912878905940866975369743071441424192833481004753922122458993040...
A7 = 1.1416118168318711437057727816148048057614284471759625288073915723140...
A8 = 0.2618221765943171424958051160111945242076019991649774700610674747694...
(End)
MATHEMATICA
Accumulate[DivisorSigma[0, Range[50]]^3]
PROG
(PARI) a(n) = sum(k=1, n, numdiv(k)^3); \\ Michel Marcus, Sep 03 2018
KEYWORD
nonn
AUTHOR
Vaclav Kotesovec, Sep 02 2018
STATUS
approved