A143127 -id:A143127 - cp's OEIS frontend

A038040 a(n) = n*d(n), where d(n) = number of divisors of n (A000005).

1, 4, 6, 12, 10, 24, 14, 32, 27, 40, 22, 72, 26, 56, 60, 80, 34, 108, 38, 120, 84, 88, 46, 192, 75, 104, 108, 168, 58, 240, 62, 192, 132, 136, 140, 324, 74, 152, 156, 320, 82, 336, 86, 264, 270, 184, 94, 480, 147, 300, 204, 312, 106, 432, 220, 448, 228, 232, 118

Offset: 1

Christian G. Bower

Dirichlet convolution of sigma(n) (A000203) with phi(n) (A000010). - Michael Somos, Jun 08 2000
Dirichlet convolution of f(n)=n with itself. See the Apostol reference for Dirichlet convolutions. - Wolfdieter Lang, Sep 09 2008
Sum of all parts of all partitions of n into equal parts. - Omar E. Pol, Jan 18 2013

			For n = 6 the partitions of 6 into equal parts are [6], [3, 3], [2, 2, 2], [1, 1, 1, 1, 1, 1]. The sum of all parts is 6 + 3 + 3 + 2 + 2 + 2 + 1 + 1 + 1 + 1 + 1 + 1 = 24 equalling 6 times the number of divisors of 6, so a(6) = 24. - _Omar E. Pol_, May 08 2021

Tom M. Apostol, Introduction to Analytic Number Theory, Springer-Verlag, 1976, pp. 29 ff.
James J. Tattersall, Elementary Number Theory in Nine Chapters, Cambridge University Press, 1999, page 162.

T. D. Noe, Table of n, a(n) for n = 1..1000
J. Bourgain, S. V. Konyagin and I. E. Shparlinski, Product sets of rationals, multiplicative translates of subgroups in residue rings and fixed points of the discrete logarithms, Int. Math. Res. Notices, 2008 (2008), Art. ID rnn 090, 1-29.
Jean Bourgain, Sergei Konyagin and Igor Shparlinski. Distribution on elements of cosets of small subgroups and applications, arXiv:1103.0567 [math.NT], Mar 2 2011.
Mikhail R. Gabdullin and Vitalii V. Iudelevich, Numbers of the form kf(k), arXiv:2201.09287 [math.NT] (2022).
Passawan Noppakaew and Prapanpong Pongsriiam, Product of Some Polynomials and Arithmetic Functions, J. Int. Seq. (2023) Vol. 26, Art. 23.9.1.
Paul Pollack, Analytic and Combinatorial Number Theory Course Notes, p. 147. [Broken link?]
Paul Pollack, Analytic and Combinatorial Number Theory Course Notes, p. 147.

Cf. A000005, A000010, A000203, A001620, A018804, A029935, A064987, A062952.
Cf. A127648, A127093, A127528.
Cf. A038044, A143127 (partial sums), A328722 (Dirichlet inverse).
Column 1 of A329323.

a038040 n = a000005 n * n  -- Reinhard Zumkeller, Jan 21 2014

with(numtheory): A038040 := n->tau(n)*n;

a[n_] := DivisorSigma[0, n]*n; Table[a[n], {n, 1, 60}] (* Jean-François Alcover, Sep 03 2012 *)

n*numlib::tau (n)$ n=1..90 // Zerinvary Lajos, May 13 2008

a(n)=if(n<1,0,direuler(p=2,n,1/(1-p*X)^2)[n])

a(n)=if(n<1,0,polcoeff(sum(k=1,n,k*x^k/(x^k-1)^2,x*O(x^n)),n)) /* Michael Somos, Jan 29 2005 */

a(n) = n*numdiv(n); \\ Michel Marcus, Oct 24 2020

from sympy import divisor_count as d
def a(n): return n*d(n)
print([a(n) for n in range(1, 60)]) # Michael S. Branicky, Mar 15 2022

[n*sigma(n,0) for n in range(1, 60)] # Stefano Spezia, Jul 20 2025

Dirichlet g.f.: zeta(s-1)^2.
G.f.: Sum_{n>=1} n*x^n/(1-x^n)^2. - Vladeta Jovovic, Dec 30 2001
Sum_{k=1..n} sigma(gcd(n, k)). Multiplicative with a(p^e) = (e+1)*p^e. - Vladeta Jovovic, Oct 30 2001
Equals A127648 * A127093 * the harmonic series, [1/1, 1/2, 1/3, ...]. - Gary W. Adamson, May 10 2007
Equals row sums of triangle A127528. - Gary W. Adamson, May 21 2007
a(n) = n*A000005(n) = A066186(n) - n*(A000041(n) - A000005(n)) = A066186(n) - n*A144300(n). - Omar E. Pol, Jan 18 2013
a(n) = A000203(n) * A240471(n) + A106315(n). - Reinhard Zumkeller, Apr 06 2014
L.g.f.: Sum_{k>=1} x^k/(1 - x^k) = Sum_{n>=1} a(n)*x^n/n. - Ilya Gutkovskiy, May 13 2017
a(n) = Sum_{d|n} A018804(d). - Amiram Eldar, Jun 23 2020
a(n) = Sum_{d|n} phi(d)*sigma(n/d). - Ridouane Oudra, Jan 21 2021
G.f.: Sum_{n >= 1} q^(n^2)*(n^2 + 2*n*q^n - n^2*q^(2*n))/(1 - q^n)^2. - Peter Bala, Jan 22 2021
a(n) = Sum_{k=1..n} sigma(n/gcd(n,k))*phi(gcd(n,k))/phi(n/gcd(n,k)). - Richard L. Ollerton, May 07 2021
Define f(x) = #{n <= x: a(n) <= x}. Gabdullin & Iudelevich show that f(x) ~ x/sqrt(log x). That is, there are 0 < A < B such that Ax/sqrt(log x) < f(x) < Bx/sqrt(log x). - Charles R Greathouse IV, Mar 15 2022
Sum_{k=1..n} a(k) ~ n^2*log(n)/2 + (gamma - 1/4)*n^2, where gamma is Euler's constant (A001620). - Amiram Eldar, Oct 25 2022
Mobius transform of A060640. - R. J. Mathar, Feb 07 2023

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

A051336 Number of increasing arithmetic progressions in {1,2,3,...,n}, including trivial arithmetic progressions of lengths 1 and 2.

Original entry on oeis.org

1, 3, 7, 13, 22, 33, 48, 65, 86, 110, 138, 168, 204, 242, 284, 330, 381, 434, 493, 554, 621, 692, 767, 844, 929, 1017, 1109, 1205, 1307, 1411, 1523, 1637, 1757, 1881, 2009, 2141, 2282, 2425, 2572, 2723, 2882, 3043, 3212, 3383, 3560, 3743, 3930, 4119

Offset: 1

John W. Layman, Nov 02 1999

The number of arithmetic subsequences of [1, ..., n] with successive-term increment i and length k is (n-i*(k-1))(i > 0, k > 0, n > i*(k-1)). - Robert E. Sawyer (rs.1(AT)mindspring.com)

The best algorithm known for generating a(n) from scratch has order O(sqrt(n)) (see below). If a(n-1) is known, it reduces to O(n^(1/3)). - Daniel Hoying, May 20 2020

			a(1): [1];
a(2): [1],[2],[1,2];
a(3): [1],[2],[3],[1,2],[1,3],[2,3],[1,2,3].

T. D. Noe, Table of n, a(n) for n = 1..1000
Marcel K. Goh, Jad Hamdan, and Jonah Saks, The lattice of arithmetic progressions, arXiv:2106.05949 [math.CO], 2021.
Daniel Hoying, Proof of recurrence relation, May 19 2020.

Cf. A078567.
Cf. A006218, A143127.
See A078651 and A366471 for GPs.

nmax = 48; t = Table[ DivisorSigma[0, n], {n, 1, nmax}]; Accumulate[ Accumulate[t]+1] - Accumulate[t] (* Jean-François Alcover, Nov 08 2011 *)
With[{c=Accumulate[DivisorSigma[0,Range[50]]]},Accumulate[c+1]-c] (* Harvey P. Dale, Dec 23 2015 *)
nmax = 50; RecurrenceTable[{a[n] == a[n-1]+1+p[n], p[n] == p[n-1]+DivisorSigma[0, n-1], a[1] == 1, p[1] == 0}, {a, p}, {n, 1, nmax}][[All,1]] (* Daniel Hoying, May 16 2020 *)

from math import isqrt
def A051336(n): return (((s:=isqrt(n-1))*(s+1))**2>>2)+(1-s**2)*n+sum((q:=(n-1)//k)*(2*n-k*(1+q)) for k in range(1, s+1)) # Chai Wah Wu, Oct 21 2023

Theorem: the second differences give tau(n+1), the number of divisors of n+1 (A000005).
a(n) = n + A078567(n).
a(n) = n + Sum_{ i=1..n-1, j=1..floor(n/i) } (n - i*j). - Robert E. Sawyer (rs.1(AT)mindspring.com)
From Daniel Hoying, May 15 2020: (Start)
a(n+1) = a(n) + 1 + Sum_{i=1..n} tau(i).
       = a(n) + 1 + A006218(n+1).
a(n+1) = (n + 1)*(1 + Sum_{i=1..n} floor(n/i)) - Sum_{i=1..n} i*tau(i).
       = (n + 1)*(1 + A006218(n)) - A143127(n). (End)

A083356 Total area of all incongruent integer-sided rectangles of area <= n.

Original entry on oeis.org

0, 1, 3, 6, 14, 19, 31, 38, 54, 72, 92, 103, 139, 152, 180, 210, 258, 275, 329, 348, 408, 450, 494, 517, 613, 663, 715, 769, 853, 882, 1002, 1033, 1129, 1195, 1263, 1333, 1513, 1550, 1626, 1704, 1864, 1905, 2073, 2116, 2248, 2383, 2475, 2522, 2762, 2860

Offset: 0

Dean Hickerson, Apr 26 2003

			a(5)=19, the rectangles being 1 X 1, 1 X 2, 1 X 3, 1 X 4, 1 X 5 and 2 X 2.

Nick MacKinnon, Problem 10883, Amer. Math. Monthly, 108 (2001) 565; solution by John C. Cock, 110 (2003) 343-344.

Cf. A083357, A143127

Partial sums of A060872.

a[n_] := Sum[r(r+Floor[n/r])(Floor[n/r]+1-r), {r, 1, Floor[Sqrt[n]]}]/2

from math import isqrt
def A083356(n): return (k:=isqrt(n))*(k+1)*(2+4*k-3*k*(k+1))//24+sum(i*(m:=n//i)*(1+m)>>1 for i in range(1,k+1)) # Chai Wah Wu, Jul 11 2023

a(n) = Sum_{k=1..n} k*ceiling(d(k)/2), where d(k)=A000005(k) is the number of divisors of k.
a(n) = Sum_{r=1..floor(sqrt(n))} r*(r+floor(n/r))*(floor(n/r)+1-r)/2.
a(n) = ( A143127(n) + A000330(floor(sqrt(n))) ) / 2. - Max Alekseyev, Jan 31 2012
a(n) ~ n^2 * log(n) / 4
G.f.: x*f'(x)/(1 - x), where f(x) = Sum_{k>=1} x^k^2/(1 - x^k). - Ilya Gutkovskiy, Apr 12 2017

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

A350107 a(n) = Sum_{k=1..n} k * floor(n/k)^2.

Original entry on oeis.org

1, 6, 14, 31, 45, 81, 101, 150, 191, 253, 285, 401, 439, 527, 623, 752, 802, 979, 1035, 1233, 1369, 1509, 1577, 1901, 2020, 2186, 2362, 2642, 2728, 3136, 3228, 3549, 3765, 3983, 4215, 4772, 4882, 5126, 5382, 5932, 6054, 6630, 6758, 7202, 7664, 7960, 8100, 8936

Offset: 1

Seiichi Manyama, Dec 14 2021

nonn

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

Column 2 of A350106.

Cf. A000005 (tau), A024916, A143127, A222548, A350123, A350128.

a[n_] := Sum[k * Floor[n/k]^2, {k, 1, n}]; Array[a, 50] (* Amiram Eldar, Dec 14 2021 *)
Accumulate[Table[2*k*DivisorSigma[0, k] - DivisorSigma[1, k], {k, 1, 100}]] (* Vaclav Kotesovec, Dec 16 2021 *)

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

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

my(N=66, x='x+O('x^N)); Vec(sum(k=1, N, (2*k-1)*x^k/(1-x^k)^2)/(1-x))

a(n) = sum(k=1, n, 2*k*numdiv(k)-sigma(k));

from math import isqrt
def A350107(n): return -(s:=isqrt(n))**3*(s+1)+sum((q:=n//k)*((k<<1)*((q<<1)+1)-q-1) for k in range(1,s+1))>>1 # Chai Wah Wu, Oct 24 2023

a(n) = Sum_{k=1..n} k * Sum_{d|k} (2*d - 1)/d = 2 * A143127(n) - A024916(n).
G.f.: (1/(1 - x)) * Sum_{k>=1} (2*k - 1) * x^k/(1 - x^k)^2.
a(n) = Sum_{k=1..n} 2 * k * tau(k) - sigma(k).

A356124 Square array T(n,k), n >= 1, k >= 0, read by antidiagonals downwards, where T(n,k) = Sum_{j=1..n} j^k * binomial(floor(n/j)+1,2).

Original entry on oeis.org

1, 1, 4, 1, 5, 8, 1, 7, 11, 15, 1, 11, 19, 23, 21, 1, 19, 41, 47, 33, 33, 1, 35, 103, 125, 77, 57, 41, 1, 67, 281, 395, 255, 149, 71, 56, 1, 131, 799, 1373, 1025, 555, 205, 103, 69, 1, 259, 2321, 5027, 4503, 2537, 905, 325, 130, 87, 1, 515, 6823, 18965, 20657, 12867, 4945, 1585, 442, 170, 99

Offset: 1

Seiichi Manyama, Jul 27 2022

			Square array begins:
   1,  1,   1,   1,    1,     1,     1, ...
   4,  5,   7,  11,   19,    35,    67, ...
   8, 11,  19,  41,  103,   281,   799, ...
  15, 23,  47, 125,  395,  1373,  5027, ...
  21, 33,  77, 255, 1025,  4503, 20657, ...
  33, 57, 149, 555, 2537, 12867, 68969, ...

Column k=0..4 give A024916, A143127, A143128, A356125, A356126.
T(n,n) gives A356129.
T(n,n+1) gives A356128.
Cf. A279394, A319649, A350106.

T[n_, k_] := Sum[j^k * Binomial[Floor[n/j] + 1, 2], {j, 1, n}]; Table[T[k, n - k], {n, 1, 11}, {k, 1, n}] // Flatten (* Amiram Eldar, Jul 28 2022 *)

T(n, k) = sum(j=1, n, j^k*binomial(n\j+1, 2));

T(n, k) = sum(j=1, n, j*sigma(j, k-1));

from itertools import count, islice
from math import isqrt
from sympy import bernoulli
def A356124_T(n,k): return ((s:=isqrt(n))*(s+1)*(bernoulli(k+1)-bernoulli(k+1,s+1))+sum(w**k*(k+1)*((q:=n//w)*(q+1))+(w*(bernoulli(k+1,q+1)-bernoulli(k+1))<<1) for w in range(1,s+1)))//(k+1)>>1
def A356124_gen(): # generator of terms
     return (A356124_T(k+1,n-k-1) for n in count(1) for k in range(n))
A356124_list = list(islice(A356124_gen(),30)) # Chai Wah Wu, Oct 24 2023

G.f. of column k: (1/(1-x)) * Sum_{j>=1} j^k * x^j/(1 - x^j)^2.

T(n,k) = Sum_{j=1..n} j * sigma_{k-1}(j).

cp's OEIS Frontend

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