A319085 -id:A319085 - cp's OEIS frontend

A034714 Dirichlet convolution of squares with themselves.

1, 8, 18, 48, 50, 144, 98, 256, 243, 400, 242, 864, 338, 784, 900, 1280, 578, 1944, 722, 2400, 1764, 1936, 1058, 4608, 1875, 2704, 2916, 4704, 1682, 7200, 1922, 6144, 4356, 4624, 4900, 11664, 2738, 5776, 6084, 12800, 3362, 14112, 3698, 11616, 12150, 8464

Offset: 1

Erich Friedman

Bruno Berselli, Table of n, a(n) for n = 1..1000
Joerg Arndt, On computing the generalized Lambert series, arXiv:1202.6525v3 [math.CA], (2012).

Cf. A000005, A000290, A001620, A038040, A134576, A319085 (partial sums).

[n^2*NumberOfDivisors(n): n in [1..50]]; // Bruno Berselli, Feb 12 2014

A034714 := proc(n) n^2*numtheory[tau](n) ; end proc:
seq(A034714(n),n=1..20) ; # R. J. Mathar, Feb 03 2011

A034714[n_]:=DivisorSigma[0,n]*n^2; Array[A034714, 50] (* Enrique Pérez Herrero, Jun 26 2011 *)

A034714(n)=numdiv(n)*n^2 \\ Enrique Pérez Herrero, Jun 26 2011

Dirichlet g.f.: zeta^2(s-2).
Equals n^2*tau(n), where tau(n) = A000005(n) = number of divisors of n. - Jon Perry, Aug 28 2005
Multiplicative with a(p^e) = (e+1)p^(2e). - Mitch Harris, Jun 27 2005
Row sums of triangle A134576. - Gary W. Adamson, Nov 02 2007
G.f.: Sum_{k>=1} k^2*x^k*(1 + x^k)/(1 - x^k)^3. - Ilya Gutkovskiy, Oct 24 2018
a(n) = n * A038040(n). - Torlach Rush, Feb 01 2019
Sum_{k>=1} 1/a(k) = Product_{primes p} (-p^2 * log(1 - 1/p^2)) = 1.27728092754165872535305748273941301416624226497497308879403022758421224... - Vaclav Kotesovec, Sep 19 2020
G.f.: Sum_{n >= 1} q^(n^2)*( n^4*q^(3*n) - n^2*(n^2 + 4*n - 2)*q^(2*n) - n^2*(n^2 - 4*n - 2)*q^n + n^4 )/(1 - q^n)^3 - apply the operator q*d/dq twice to equation 5 in Arndt and set x = 1. - Peter Bala, Jan 21 2021
Sum_{k=1..n} a(k) ~ (n^3/3) * (log(n) + 2*gamma - 1/3), where gamma is Euler's constant (A001620). - Amiram Eldar, Nov 02 2023
a(n) = Sum_{1 <= i, j <= n} sigma_2( gcd(i, j, n) ) = Sum_{d divides n} sigma_2(d) * J_2(n/d), where sigma_2(n) = A001157(n) and the Jordan totient function J_2(n) = A007434(n). - Peter Bala, Jan 22 2024

A320895 a(n) = Sum_{k=1..n} k^3 * tau(k), where tau is A000005.

Original entry on oeis.org

1, 17, 71, 263, 513, 1377, 2063, 4111, 6298, 10298, 12960, 23328, 27722, 38698, 52198, 72678, 82504, 117496, 131214, 179214, 216258, 258850, 283184, 393776, 440651, 510955, 589687, 721399, 770177, 986177, 1045759, 1242367, 1386115, 1543331, 1714831, 2134735

Offset: 1

Vaclav Kotesovec, Oct 23 2018

nonn

In general, for m>=0, Sum_{k=1..n} k^m * tau(k) ~ n^(m+1) * ((log(n) + 2*gamma)/(m+1) - 1/(m+1)^2), where gamma is the Euler-Mascheroni constant A001620.

Vaclav Kotesovec, Table of n, a(n) for n = 1..10000
Vaclav Kotesovec, Graph - The asymptotic ratio (1000000 terms)

Cf. A000005, A006218, A143127, A319085.

Accumulate[Table[k^3*DivisorSigma[0, k], {k, 1, 50}]]

a(n) = sum(k=1, n, k^3*numdiv(k)); \\ Michel Marcus, Oct 23 2018

a(n) ~ n^4 * (log(n) + 2*gamma - 1/4)/4, where gamma is the Euler-Mascheroni constant A001620.

a(n) = Sum_{k=1..n} (k^3 / 4) * floor(n/k)^2 * floor(1 + n/k)^2. - Daniel Suteu, Nov 07 2018

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

Original entry on oeis.org

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)

A106846 a(n) = Sum_{k + lm <= n} (k + lm), with 0 <= k,l,m <= n.

Original entry on oeis.org

0, 4, 22, 64, 144, 269, 461, 720, 1072, 1522, 2092, 2774, 3626, 4614, 5776, 7126, 8694, 10445, 12461, 14684, 17204, 19997, 23077, 26412, 30156, 34206, 38600, 43352, 48532, 54042, 60072, 66458, 73338, 80664, 88450, 96710, 105638, 114999

Offset: 0

Ralf Stephan, May 06 2005

nonn

R. J. Mathar, Table of n, a(n) for n = 0..1000

Cf. A106633, A106634, A106847.

Cf. A055112, A006218, A143127, A319085, A059270, A143127.

A106846 := proc(n)
    local a,k,l,m ;
    a := 0 ;
    for k from 0 to n do
        for l from 0 to n do
            if l = 0 then
                m := n ;
            else
                m := floor((n-k)/l) ;
            end if;
            if m >=0 then
                m := min(m,n) ;
                a := a+(m+1)*k+l*m*(m+1)/2 ;
            end if;
        end do:
    end do:
    a ;
end proc: # R. J. Mathar, Oct 17 2012

A106846[n_] := Module[{a, k, l, m }, a = 0; For[k = 0, k <= n, k++, For[l = 0, l <= n, l++, If[l == 0, m = n, m = Floor[(n - k)/l]]; If[m >= 0, m = Min[m, n]; a = a + (m + 1)*k + l*m*(m + 1)/2 ]]]; a];
Table[A106846[n], {n, 0, 40}] (* Jean-François Alcover, Apr 04 2024, after R. J. Mathar *)

From Ridouane Oudra, Jun 24 2024: (Start)
a(n) = (1/2) * (n*(n+1)*(2*n+1) + Sum_{k=1..n} (n^2 + n + k - k^2) * tau(k)).
a(n) = (1/2) * (A055112(n) + (n^2 + n) * A006218(n) + A143127(n) - A319085(n)).
a(n) = A059270(n) + A143127(n) + A106847(n). (End)

A106847 a(n) = Sum {k + jm <= n} (k + jm), with 0 < k,j,m <= n.

Original entry on oeis.org

0, 0, 2, 11, 31, 71, 131, 229, 357, 537, 767, 1064, 1412, 1867, 2385, 3000, 3720, 4570, 5506, 6608, 7808, 9194, 10734, 12436, 14260, 16360, 18622, 21079, 23739, 26668, 29758, 33199, 36815, 40742, 44924, 49369, 54085, 59265, 64661, 70355

Offset: 0

Ralf Stephan, May 06 2005

nonn

			We have 1+1*1=2<=3, 1+2*1=3, 1+1*2=3, 2+1*1=3, thus a(3)=2+3+3+3=11.

M. F. Hasler, Table of n, a(n) for n = 0..9999

Cf. A106633, A106634, A106846, A078567 (number of terms).

Cf. A006218, A143272, A143274, A143127, A319085.

A106847 := proc(n)
    local a,k,l,m ;
    a := 0 ;
    for k from 1 to n do
        for l from 1 to n-k do
            m := floor((n-k)/l) ;
            if m >=1 then
                m := min(m,n) ;
                a := a+m*k+l*m*(m+1)/2 ;
            end if;
        end do:
    end do:
    a ;
end proc: # R. J. Mathar, Oct 17 2012

A106847[n_] := Module[{a, k, l, m}, a = 0; For[k = 1, k <= n, k++, For[l = 1, l <= n - k, l++, If[l == 0, m = n, m = Floor[(n - k)/l]]; If[m >= 1, m = Min[m, n]; a = a + m*k + l*m*(m + 1)/2]]]; a];
Table[A106847[n], {n, 0, 40}] (* Jean-François Alcover, Apr 04 2024, after R. J. Mathar *)

A106847(n)=sum(m=1,n-1,sum(k=1,(n-1)\m,(n-m*k)*(n+m*k+1)))/2  \\ M. F. Hasler, Oct 17 2012

From Ridouane Oudra, Jun 02 2024: (Start)
a(n) = (1/2)*Sum_{k=1..n} (n^2 + n - k^2 - k)*tau(k);
a(n) = (1/2)*(n^2 + n)*A006218(n) - Sum_{k=1..n} A143272(k);
a(n) = (1/2)*((n + 1)*A143274(n) - A143127(n) - A319085(n)). (End)
a(n) ~ n^3 * (log(n) + 2*gamma - 4/3)/3, where gamma is the Euler-Mascheroni constant A001620. - Vaclav Kotesovec, Jun 15 2024

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.

A350124 a(n) = Sum_{k=1..n} k^2 * floor(n/k)^3.

Original entry on oeis.org

1, 12, 40, 121, 207, 473, 649, 1142, 1611, 2401, 2853, 4647, 5285, 6879, 8759, 11452, 12558, 16739, 18127, 23353, 27129, 31171, 33219, 43573, 47524, 53210, 59538, 69996, 73274, 89694, 93446, 107195, 116731, 126545, 137505, 164580, 169946, 182244, 195644, 225454

Offset: 1

Seiichi Manyama, Dec 15 2021

nonn

Seiichi Manyama, Table of n, a(n) for n = 1..10000

Cf. A318742, A350108, A350123, A350125.

Cf. A064602, A143128, A319085.

Accumulate[Table[DivisorSigma[2, k] - 3*k*DivisorSigma[1, k] + 3*k^2*DivisorSigma[0, k], {k, 1, 50}]] (* Vaclav Kotesovec, Dec 17 2021 *)

PARI
```
a(n) = sum(k=1, n, k^2*(n\k)^3);
```

a(n) = sum(k=1, n, k^2*sumdiv(k, d, (d^3-(d-1)^3)/d^2));

my(N=66, x='x+O('x^N)); Vec(sum(k=1, N, (k^3-(k-1)^3)*x^k*(1+x^k)/(1-x^k)^3)/(1-x))

from math import isqrt
def A350124(n): return (-(s:=isqrt(n))**4*(s+1)*(2*s+1) + sum((q:=n//k)*(k*(3*(k-1))+q*(k*(9*(k-1))+q*(k*(12*k-6)+2)+3)+1) for k in range(1,s+1)))//6 # Chai Wah Wu, Oct 21 2023

a(n) = Sum_{k=1..n} k^2 * Sum_{d|k} (d^3 - (d - 1)^3)/d^2.
G.f.: (1/(1 - x)) * Sum_{k>=1} (k^3 - (k - 1)^3) * x^k * (1 + x^k)/(1 - x^k)^3.
From Vaclav Kotesovec, Aug 03 2022: (Start)
a(n) = A064602(n) - 3*A143128(n) + 3*A319085(n).
a(n) ~ n^3 * (log(n) + 2*gamma + (zeta(3) - 1)/3 - Pi^2/6), where gamma is the Euler-Mascheroni constant A001620. (End)

cp's OEIS Frontend

A034714 Dirichlet convolution of squares with themselves.

Views

Author

Keywords

Links

Crossrefs

Programs

Magma

Maple

Mathematica

PARI

Formula

A320895 a(n) = Sum_{k=1..n} k^3 * tau(k), where tau is A000005.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

PARI

Formula

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

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A106846 a(n) = Sum_{k + l*m <= n} (k + l*m), with 0 <= k,l,m <= n.

Views

Author

Keywords

Links

Crossrefs

Programs

Maple

Mathematica

Formula

A106847 a(n) = Sum {k + j*m <= n} (k + j*m), with 0 < k,j,m <= n.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Maple

Mathematica

PARI

Formula

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

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

PARI

Formula

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

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A350124 a(n) = Sum_{k=1..n} k^2 * floor(n/k)^3.

Views

Author

A106846 a(n) = Sum_{k + lm <= n} (k + lm), with 0 <= k,l,m <= n.

A106847 a(n) = Sum {k + jm <= n} (k + jm), with 0 < k,j,m <= n.