A320895 -id:A320895 - cp's OEIS frontend

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

Vaclav Kotesovec, Sep 02 2018

nonn

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.

Cf. A000005, A006218, A061502, A319089.

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

Accumulate[DivisorSigma[0, Range[50]]^3]

a(n) = sum(k=1, n, numdiv(k)^3); \\ Michel Marcus, Sep 03 2018

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)

A320896 a(n) = Sum_{k=1..n} k * tau(k)^2, where tau is A000005.

Original entry on oeis.org

1, 9, 21, 57, 77, 173, 201, 329, 410, 570, 614, 1046, 1098, 1322, 1562, 1962, 2030, 2678, 2754, 3474, 3810, 4162, 4254, 5790, 6015, 6431, 6863, 7871, 7987, 9907, 10031, 11183, 11711, 12255, 12815, 15731, 15879, 16487, 17111, 19671, 19835, 22523, 22695, 24279

Offset: 1

Vaclav Kotesovec, Oct 23 2018

nonn

Amiram Eldar, Table of n, a(n) for n = 1..10000
Ramanujan's Papers, Some formulas in the analytic theory of numbers, Messenger of Mathematics, XLV, 1916, 81-84.

Cf. A061502, A318755, A320897.

Cf. A000005, A006218, A143127, A319085, A320895.

Accumulate[Table[k*DivisorSigma[0, k]^2, {k, 1, 50}]]

a(n) = sum(k=1, n, k*numdiv(k)^2); \\ Michel Marcus, Oct 23 2018

a(n) ~ n^2 * (3*(Pi^6*(-1 - 24*g^2 + 32*g^3 + g*(8 - 96*s1) + 16*s1 + 16*s2) - 13824*z1^3 + 576*Pi^2*z1*((-1 + 8*g)*z1 + 4*z2) - 8*Pi^4*(3*(1 - 8*g + 24*g^2 - 16*s1)*z1 - 6*z2 + 48*g*z2 + 8*z3)) + 6*(Pi^6*(1 - 8*g + 24*g^2 - 16*s1) + 576*Pi^2*z1^2 - 24*Pi^4*(-z1 + 8*g*z1 + 2*z2))*log(n) + 6*((-1 + 8*g)*Pi^6 - 24*Pi^4*z1)*log(n)^2 + 4*Pi^6*log(n)^3) / (8*Pi^8), where g is the Euler-Mascheroni constant A001620, z1 = Zeta'(2) = A073002, z2 = Zeta''(2) = A201994, z3 = Zeta'''(2) = A201995 and s1, s2 are the Stieltjes constants, see A082633 and A086279.

A320897 a(n) = Sum_{k=1..n} k^2 * tau(k)^2, where tau is A000005.

Original entry on oeis.org

1, 17, 53, 197, 297, 873, 1069, 2093, 2822, 4422, 4906, 10090, 10766, 13902, 17502, 23902, 25058, 36722, 38166, 52566, 59622, 67366, 69482, 106346, 111971, 122787, 134451, 162675, 166039, 223639, 227483, 264347, 281771, 300267, 319867, 424843, 430319, 453423

Offset: 1

Vaclav Kotesovec, Oct 23 2018

nonn

In general, for m>=0, Sum_{k=1..n} k^m * tau(k)^2 ~ n^(m+1) * (log(n))^3 / ((m+1) * Pi^2).

Amiram Eldar, Table of n, a(n) for n = 1..10000
Ramanujan's Papers, Some formulas in the analytic theory of numbers, Messenger of Mathematics, XLV, 1916, 81-84.

Cf. A061502, A318755, A320896.

Cf. A000005, A006218, A143127, A319085, A320895.

Accumulate[Table[k^2*DivisorSigma[0, k]^2, {k, 1, 50}]]

a(n) = sum(k=1, n, k^2*numdiv(k)^2); \\ Michel Marcus, Oct 23 2018

a(n) ~ n^3 * (2*Pi^6*(-1 + 12*g - 54*g^2 + 108*g^3 + 36*s1 - 324*g*s1 + 54*s2) - 93312*z1^3 + 2592*Pi^2*z1*(-z1 + 12*g*z1 + 6*z2) - 72*Pi^4*(z1 - 12*g*z1 + 54*g^2*z1 - 36*s1*z1 - 3*z2 + 36*g*z2 + 6*z3) + 6*(Pi^6*(1 - 12*g + 54*g^2 - 36*s1) + 1296*Pi^2*z1^2 - 36*Pi^4*(-z1 + 12*g*z1 + 3*z2))*log(n) + 9*((-1 + 12*g)*Pi^6 - 36*Pi^4*z1)*log(n)^2 + 9*Pi^6*log(n)^3) / (27*Pi^8), where g is the Euler-Mascheroni constant A001620, z1 = Zeta'(2) = A073002, z2 = Zeta''(2) = A201994, z3 = Zeta'''(2) = A201995 and s1, s2 are the Stieltjes constants, see A082633 and A086279.

A386012 a(n) = n^3*tau(n).

Original entry on oeis.org

1, 16, 54, 192, 250, 864, 686, 2048, 2187, 4000, 2662, 10368, 4394, 10976, 13500, 20480, 9826, 34992, 13718, 48000, 37044, 42592, 24334, 110592, 46875, 70304, 78732, 131712, 48778, 216000, 59582, 196608, 143748, 157216, 171500, 419904, 101306, 219488, 237276, 512000

Offset: 1

R. J. Mathar, Jul 14 2025

Dirichlet convolution of the cubes A000578 with themselves.

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A000005, A001620, A034714, A038040, A320895 (partial sums), A372928 (Mobius transform).

Maple
```
seq( n^3*numtheory[tau](n),n=1..100) ;
```

a[n_]:=n^3*DivisorSigma[0,n]; Array[a,40] (* Stefano Spezia, Jul 14 2025 *)
nmax = 40; Rest[CoefficientList[Series[Sum[k^3*x^k*(1 + 4*x^k + x^(2*k))/(1 - x^k)^4, {k, 1, nmax}], {x, 0, nmax}], x]] (* Vaclav Kotesovec, Aug 03 2025 *)

a(n) = n^3 * numdiv(n); \\ Amiram Eldar, Jul 15 2025

a(n) = n*A034714(n) = n^2*A038040(n).
Dirichlet g.f.: zeta^2(s-3).
From Amiram Eldar, Jul 15 2025 (Start)
Multiplicative with a(p^e) = p^(3*e) * (e+1).
Sum_{k=1..n} a(k) ~ (n^4/4) * (log(n) + 2*gamma - 1/4), where gamma is Euler's constant (A001620). (End)
G.f.: Sum_{k>=1} k^3*x^k*(1 + 4*x^k + x^(2*k)) / (1-x^k)^4. - Vaclav Kotesovec, Aug 03 2025

A364194 a(n) = Sum_{k=1..n} k^3*sigma(k), where sigma is A000203.

Original entry on oeis.org

1, 25, 133, 581, 1331, 3923, 6667, 14347, 23824, 41824, 57796, 106180, 136938, 202794, 283794, 410770, 499204, 726652, 863832, 1199832, 1496184, 1879512, 2171520, 3000960, 3485335, 4223527, 5010847, 6240159, 6971829, 8915829, 9869141, 11933525, 13658501

Offset: 1

Seiichi Manyama, Oct 20 2023

Wikipedia, Faulhaber's formula.

Partial sums of A282211.
Cf. A319086, A320895, A356125, A364269.
Cf. A000203, A000537, A143128.

Accumulate[Table[n^3*DivisorSigma[1, n], {n, 1, 33}]] (* Amiram Eldar, Oct 20 2023 *)

f(n, m) = (subst(bernpol(m+1, x), x, n+1)-subst(bernpol(m+1, x), x, 0))/(m+1);
a(n, s=3, t=1) = sum(k=1, n, k^(s+t)*f(n\k, s));

def A364194(n): return sum((k**2*(m:=n//k)*(m+1)>>1)**2 for k in range(1,n+1)) # Chai Wah Wu, Oct 20 2023

from math import isqrt
def A364194(n): return ((((s:=isqrt(n))*(s + 1))**3*(2*s+1)*(1-3*s*(s+1))>>1) + sum((q:=n//k)*(q+1)*k**3*(q*(15*k+q*(15*k+12*q+18)+2)-2) for k in range(1,s+1)))//60 # Chai Wah Wu, Oct 21 2023

a(n) = Sum_{k=1..n} k^4 * A000537(floor(n/k)).

a(n) ~ (zeta(2)/5) * n^5. - Amiram Eldar, Oct 20 2023

cp's OEIS Frontend

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

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A320896 a(n) = Sum_{k=1..n} k * tau(k)^2, where tau is A000005.

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A320897 a(n) = Sum_{k=1..n} k^2 * tau(k)^2, where tau is A000005.

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A386012 a(n) = n^3*tau(n).

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A364194 a(n) = Sum_{k=1..n} k^3*sigma(k), where sigma is A000203.

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