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A318755
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a(n) = Sum_{k=1..n} tau(k)^3, where tau is A000005.
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5
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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
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OFFSET
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1,2
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LINKS
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FORMULA
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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)].
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)
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MATHEMATICA
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Accumulate[DivisorSigma[0, Range[50]]^3]
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PROG
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(PARI) a(n) = sum(k=1, n, numdiv(k)^3); \\ Michel Marcus, Sep 03 2018
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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