A034714 -id:A034714 - cp's OEIS frontend

A360429 Inverse Mobius transformation of A034714.

1, 9, 19, 57, 51, 171, 99, 313, 262, 459, 243, 1083, 339, 891, 969, 1593, 579, 2358, 723, 2907, 1881, 2187, 1059, 5947, 1926, 3051, 3178, 5643, 1683, 8721, 1923, 7737, 4617, 5211, 5049, 14934, 2739, 6507, 6441, 15963, 3363, 16929, 3699, 13851, 13362, 9531, 4419, 30267, 7302, 17334

Offset: 1

R. J. Mathar, Feb 07 2023

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

Cf. A000005, A000012, A001620, A034714, A060640.

A360429 := proc(n)
    add(numtheory[tau](d)*d^2,d=numtheory[divisors](n)) ;
end proc:

f[p_, e_] := ((e+1)*p^(2*e+4) - (e+2)*p^(2*e+2) + 1)/(p^2-1)^2; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Feb 09 2023 *)

a(n) = Sum_{d|n} A000005(d)*d^2.
Dirichlet convolution of A034714 and A000012.
Dirichlet g.f.: zeta^2(s-2)*zeta(s).
From Amiram Eldar, Feb 09 2023: (Start)
Multiplicative with a(p^e) = ((e+1)*p^(2*e+4) - (e+2)*p^(2*e+2) + 1)/(p^2-1)^2.
Sum_{k=1..n} a(k) ~ (log(n) + 2*gamma - 1/3 + zeta'(3)/zeta(3)) * n^3 * zeta(3)/3, where gamma is Euler's constant (A001620). (End)

A319085 a(n) = Sum_{k=1..n} k^2*tau(k), where tau is A000005.

Original entry on oeis.org

1, 9, 27, 75, 125, 269, 367, 623, 866, 1266, 1508, 2372, 2710, 3494, 4394, 5674, 6252, 8196, 8918, 11318, 13082, 15018, 16076, 20684, 22559, 25263, 28179, 32883, 34565, 41765, 43687, 49831, 54187, 58811, 63711, 75375, 78113, 83889, 89973, 102773, 106135

Offset: 1

Vaclav Kotesovec, Sep 10 2018

nonn

In general, for m>=1, Sum_{k=1..n} k^m * tau(k) = Sum_{k=1..n} k^m * (Bernoulli(m+1, floor(1 + n/k)) - Bernoulli(m+1, 0)) / (m+1), where Bernoulli(n,x) are the Bernoulli polynomials. - Daniel Suteu, Nov 08 2018

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

Cf. A000005, A006218, A034714, A143127.

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

a(n) = sum(k=1, n, k^2*numdiv(k)); \\ Michel Marcus, Sep 12 2018

f(n) = n*(n+1)*(2*n+1)/6; \\ A000330
a(n) = 2*sum(k=1, sqrtint(n), k^2 * f(n\k)) - f(sqrtint(n))^2; \\ Daniel Suteu, Nov 26 2020

from math import isqrt
def A319085(n): return (-((s:=isqrt(n))*(s+1)*(2*s+1))**2//12 + sum(k**2*(q:=n//k)*(q+1)*(2*q+1) for k in range(1,s+1)))//3 # Chai Wah Wu, Oct 21 2023

a(n) ~ n^3 * (log(n) + 2*gamma - 1/3)/3, where gamma is the Euler-Mascheroni constant A001620.
a(n) = Sum_{k=1..n} k^2 * Bernoulli(3, floor(1 + n/k)) / 3, where Bernoulli(n,x) are the Bernoulli polynomials. - Daniel Suteu, Nov 08 2018
a(n) = Sum_{k=1..n} Sum_{i=1..floor(n/k)} i^2 * k^2. - Wesley Ivan Hurt, Nov 26 2020

A360428 Inverse Mobius transformation of A338164.

Original entry on oeis.org

1, 7, 17, 40, 49, 119, 97, 208, 225, 343, 241, 680, 337, 679, 833, 1024, 577, 1575, 721, 1960, 1649, 1687, 1057, 3536, 1825, 2359, 2673, 3880, 1681, 5831, 1921, 4864, 4097, 4039, 4753, 9000, 2737, 5047, 5729, 10192, 3361, 11543, 3697, 9640, 11025, 7399, 4417, 17408, 7105, 12775

Offset: 1

R. J. Mathar, Feb 07 2023

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

Cf. A000005, A001620, A007434, A008683, A018804, A069097, A338164.

A360428 := proc(n)
    add(numtheory[mobius](n/d)*numtheory[tau](d)*d^2,d=numtheory[divisors](n)) ;
end proc:

f[p_, e_] := (e + 1 - e/p^2)*p^(2*e); a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Feb 09 2023 *)

a(n) = Sum_{d|n} A008683(n/d)*A000005(d)*d^2.
Dirichlet convolution of A034714 and A008683.
Dirichlet g.f.: zeta^2(s-2)/zeta(s).
From Amiram Eldar, Feb 09 2023: (Start)
Multiplicative with a(p^e) = (e + 1 - e/p^2)*p^(2*e).
Sum_{k=1..n} a(k) ~ (log(n) + 2*gamma - 1/3 - zeta'(3)/zeta(3)) * n^3 / (3*zeta(3)), where gamma is Euler's constant (A001620). (End)
From Peter Bala, Jan 16 2024: (Start)
a(n) = Sum_{1 <= i, j <= n} gcd(i, j, n)^2. Cf. A069097.
a(n) = Sum_{d divides n} d^2 * J_2(n/d), where J_2(n) = A007434(n). (End)

A309732 Expansion of Sum_{k>=1} k^2 * x^k/(1 - x^k)^3.

Original entry on oeis.org

1, 7, 15, 38, 40, 108, 77, 188, 180, 290, 187, 600, 260, 560, 630, 888, 442, 1323, 551, 1620, 1218, 1364, 805, 3024, 1325, 1898, 1998, 3136, 1276, 4680, 1457, 4080, 2970, 3230, 3290, 7470, 2072, 4028, 4134, 8200, 2542, 9072, 2795, 7656, 7830, 5888, 3337, 14496, 4998, 9825, 7038

Offset: 1

Ilya Gutkovskiy, Aug 14 2019

nonn

Dirichlet convolution of triangular numbers (A000217) with squares (A000290).

a(n) is n times half m, where m is the sum of all parts plus the total number of parts of the partitions of n into equal parts. - Omar E. Pol, Nov 30 2019

Marius A. Burtea, Table of n, a(n) for n = 1..10000

Cf. A000005, A000203, A000217, A000290, A007437, A034714, A034715, A038040, A064987, A152211, A309731, A244051.

[n*(n*NumberOfDivisors(n) + DivisorSigma(1,n))/2:n in [1..51]]; // Marius A. Burtea, Nov 29 2019

with(numtheory): seq(n*(n*tau(n)+sigma(n))/2, n=1..50); # Ridouane Oudra, Nov 28 2019

nmax = 51; CoefficientList[Series[Sum[k^2 x^k/(1 - x^k)^3, {k, 1, nmax}], {x, 0, nmax}], x] // Rest
Table[DirichletConvolve[j (j + 1)/2, j^2, j, n], {n, 1, 51}]
Table[n (n DivisorSigma[0, n] + DivisorSigma[1, n])/2, {n, 1, 51}]

a(n)=sumdiv(n, d, binomial(n/d+1,2)*d^2); \\ Andrew Howroyd, Aug 14 2019

a(n)=n*(n*numdiv(n) + sigma(n))/2; \\ Andrew Howroyd, Aug 14 2019

G.f.: Sum_{k>=1} (k*(k + 1)/2) * x^k * (1 + x^k)/(1 - x^k)^3.
a(n) = n * (n * d(n) + sigma(n))/2.
Dirichlet g.f.: zeta(s-2) * (zeta(s-2) + zeta(s-1))/2.
a(n) = n*(A038040(n) + A000203(n))/2 = n*A152211(n)/2. - Omar E. Pol, Aug 17 2019
a(n) = Sum_{k=1..n} k*sigma(gcd(n,k)). - Ridouane Oudra, Nov 28 2019

A338164 Dirichlet g.f.: (zeta(s-2) / zeta(s))^2.

Original entry on oeis.org

1, 6, 16, 33, 48, 96, 96, 168, 208, 288, 240, 528, 336, 576, 768, 816, 576, 1248, 720, 1584, 1536, 1440, 1056, 2688, 1776, 2016, 2448, 3168, 1680, 4608, 1920, 3840, 3840, 3456, 4608, 6864, 2736, 4320, 5376, 8064, 3360, 9216, 3696, 7920, 9984, 6336, 4416, 13056, 7008, 10656

Offset: 1

Ilya Gutkovskiy, Oct 14 2020

Dirichlet convolution of Jordan function J_2 (A007434) with itself.

Wikipedia, Jordan's totient function

Cf. A000005, A007427, A007434, A029935, A034714, A306379, A321322, A338165.

Jordan2[n_] := Sum[d^2 MoebiusMu[n/d], {d, Divisors[n]}]; a[n_] := Sum[Jordan2[d] Jordan2[n/d], {d, Divisors[n]}]; Table[a[n], {n, 1, 50}]
a[1] = 1; f[p_, e_] := p^(2 e - 4) (p^4 + e (p^2 - 1)^2 - 1); a[n_] := Times @@ f @@@ FactorInteger[n]; Table[a[n], {n, 1, 50}]

Multiplicative with a(p^e) = p^(2*e - 4) * (p^4 + e * (p^2 - 1)^2 - 1).
a(n) = Sum_{d|n} J_2(d) * J_2(n/d).
a(n) = Sum_{d|n} d^2 * tau(d) * A007427(n/d), where tau = A000005.
a(n) = Sum_{d|n} d^2 * A321322(n/d).
(1/tau(n)) * Sum_{d|n} a(d) * tau(n/d) = n^2.
Sum_{k=1..n} a(k) ~ ((3*log(n) + 6*gamma - 1)/(9*zeta(3)^2) - 2*zeta'(3) / (3*zeta(3)^3)) * n^3, where gamma is the Euler-Mascheroni constant A001620. - Vaclav Kotesovec, Oct 14 2020

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

A134576 A127733 * A051731.

Original entry on oeis.org

1, 4, 4, 9, 0, 9, 16, 16, 0, 16, 25, 0, 0, 0, 25, 36, 36, 36, 0, 0, 36, 49, 0, 0, 0, 0, 0, 49, 64, 64, 0, 64, 0, 0, 0, 64, 81, 0, 81, 0, 0, 0, 0, 0, 81, 100, 100, 0, 0, 100, 0, 0, 0, 0, 100

Offset: 1

Gary W. Adamson, Nov 02 2007

Row sums = A034714: (1, 8, 18, 48, 50, 144, ...).

			First few rows of the triangle:
   1;
   4,  4;
   9,  0,  9;
  16, 16,  0, 16;
  25,  0,  0,  0, 25;
  36, 36, 36,  0,  0, 36;
  49,  0,  0,  0,  0,  0, 49;
  ...

Cf. A051731, A127733, A034714.

A127733 * A051731 as infinite lower triangular matrices.

Triangle read by rows: replace 1's in n-th row of A051731 with n^2.

A328490 Dirichlet g.f.: zeta(s)^2 * zeta(s-2)^2.

Original entry on oeis.org

1, 10, 20, 67, 52, 200, 100, 380, 282, 520, 244, 1340, 340, 1000, 1040, 1973, 580, 2820, 724, 3484, 2000, 2440, 1060, 7600, 1978, 3400, 3460, 6700, 1684, 10400, 1924, 9710, 4880, 5800, 5200, 18894, 2740, 7240, 6800, 19760, 3364, 20000, 3700, 16348, 14664

Offset: 1

Ilya Gutkovskiy, Oct 16 2019

Dirichlet convolution of A001157 with itself.
Dirichlet convolution of A000005 with A034714.
Dirichlet convolution of A000290 with A007433.

Cf. A000005, A000290, A001157, A001620, A007426, A007433, A034714, A034761.

Cf. A175705.

[&+[DivisorSigma(2,d)*DivisorSigma(2, n div d):d in Divisors(n)]:n in [1..50]]; // Marius A. Burtea, Oct 16 2019

Table[DivisorSum[n, DivisorSigma[2, #] DivisorSigma[2, n/#] &], {n, 1, 45}]
f[p_, e_] :=((e*(p^2-1)+p^2-3)*p^(2*e+4) + e*(p^2-1) + 3*p^2 - 1)/(p^2-1)^3 ; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Sep 15 2023 *)

for(n=1, 100, print1(direuler(p=2, n, 1/(1 - X)^2 / (1 - p^2*X)^2)[n], ", ")) \\ Vaclav Kotesovec, Sep 26 2020

a(n) = Sum_{d|n} sigma_2(d) * sigma_2(n/d), where sigma_2 = A001157.
a(n) = Sum_{d|n} d^2 * tau(d) * tau(n/d), where tau = A000005.
Sum_{k=1..n} a(k) ~ zeta(3) * n^3 * (zeta(3)*(log(n)/3 + 2*gamma/3 - 1/9) + 2*zeta'(3)/3), where gamma is the Euler-Mascheroni constant A001620. - Vaclav Kotesovec, Oct 17 2019
Multiplicative with a(p^e) = ((e*(p^2-1)+p^2-3)*p^(2*e+4) + e*(p^2-1) + 3*p^2 - 1)/(p^2-1)^3. - Amiram Eldar, Sep 15 2023

A386013 a(n) = n^4*tau(n).

Original entry on oeis.org

1, 32, 162, 768, 1250, 5184, 4802, 16384, 19683, 40000, 29282, 124416, 57122, 153664, 202500, 327680, 167042, 629856, 260642, 960000, 777924, 937024, 559682, 2654208, 1171875, 1827904, 2125764, 3687936, 1414562, 6480000, 1847042, 6291456, 4743684, 5345344, 6002500, 15116544, 3748322, 8340544, 9253764, 20480000

Offset: 1

R. J. Mathar, Jul 14 2025

Dirichlet convolution of the 4th powers A000583 with themselves.

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

Cf. A000005, A000583, A001620, A034714, A038040, A372931 (Mobius transform).

Maple
```
seq( n^4*numtheory[tau](n),n=1..100) ;
```

a[n_]:=n^4*DivisorSigma[0,n]; Array[a,40] (* Stefano Spezia, Jul 14 2025 *)
nmax = 40; Rest[CoefficientList[Series[Sum[k^4*x^k*(1 + 11*x^k + 11*x^(2*k) + x^(3*k))/(1 - x^k)^5, {k, 1, nmax}], {x, 0, nmax}], x]] (* Vaclav Kotesovec, Aug 03 2025 *)

a(n) = n^4 * numdiv(n); \\ Amiram Eldar, Jul 15 2025

a(n) = A000005(n) * A000583(n).
a(n) = n^2*A034714(n) = n^3*A038040(n) = n*A386012(n).
Dirichlet g.f.: zeta^2(s-4).
From Amiram Eldar, Jul 15 2025 (Start)
Multiplicative with a(p^e) = p^(4*e) * (e+1).
Sum_{k=1..n} a(k) ~ (n^5/5) * (log(n) + 2*gamma - 1/5), where gamma is Euler's constant (A001620). (End)
G.f.: Sum_{k>=1} k^4*x^k*(1 + 11*x^k + 11*x^(2*k) + x^(3*k))/(1 - x^k)^5. - Vaclav Kotesovec, Aug 03 2025

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A360429 Inverse Mobius transformation of A034714.

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A319085 a(n) = Sum_{k=1..n} k^2*tau(k), where tau is A000005.

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A360428 Inverse Mobius transformation of A338164.

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A309732 Expansion of Sum_{k>=1} k^2 * x^k/(1 - x^k)^3.

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A338164 Dirichlet g.f.: (zeta(s-2) / zeta(s))^2.

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

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A134576 A127733 * A051731.

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A328490 Dirichlet g.f.: zeta(s)^2 * zeta(s-2)^2.

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