A318755 a(n) = Sum_{k=1..n} tau(k)^3, where tau is A000005.

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

Vaclav Kotesovec, Table of n, a(n) for n = 1..10000

Vaclav Kotesovec, Graph - the asymptotic ratio (100000 terms)

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

Crossrefs

Cf. A000005, A006218, A061502, A319089.

Cf. A143127, A319085, A320895, A320896, A320897.

Sequence in context: A186298 A147245 A146625 * A146576 A147138 A282764

Adjacent sequences: A318752 A318753 A318754 * A318756 A318757 A318758

Keyword

nonn

Author

Vaclav Kotesovec, Sep 02 2018

Status

approved