A000385 -id:A000385 - cp's OEIS frontend

A218278 Convolution of level 4 of the divisor function.

0, 0, 0, 0, 1, 3, 4, 7, 9, 21, 20, 36, 35, 66, 52, 101, 84, 147, 120, 224, 160, 285, 220, 394, 281, 483, 360, 680, 455, 750, 560, 1025, 680, 1116, 800, 1512, 969, 1575, 1148, 2088, 1330, 2160, 1540, 2860, 1771, 2838, 2024, 3734, 2286, 3651, 2640, 4816, 2925

Offset: 1

Michel Marcus, Oct 25 2012

nonn

Named W4(n) by S. Alaca and K. S. Williams.

S. Alaca and K. S. Williams, Evaluation of the convolution sums ..., Journal of Number Theory, Volume 124, Issue 2, June 2007, Pages 491-510.
E. Royer, Evaluating convolutions of divisor sums with quasimodular forms, arXiv:math/0510429 [math.NT], 2005-2006; International Journal of Number Theory 3, 2 (2007) p. 231-261.

Cf. A000385, A218276, A218277.

with(numtheory): seq(add(sigma(k)*sigma(n-4*k), k=1..floor(n/4)), n=1..70); # Ridouane Oudra, Nov 23 2022

a(n) = {for (i=1, n, s = sum(m=1, floor((i-1)/4), sigma(m)*sigma(i-4*m));print1(s , ", "););}

a(n) = {for (i=1, n, v = sigma(i,3)/48 - i*sigma(i)/16 + sigma(i)/24;if (i%4 == 0, v += sigma(i/4,3)/3 - i*sigma(i/4)/4 + sigma(i/4)/24);if (i%2 == 0, v += sigma(i/2,3)/16);print1(v , ", "););}

a(n) = Sum_{m<4n} sigma(n)*sigma(n-4*m).
a(n) = sigma_3(n)/48 - n*sigma(n)/16 + sigma(n)/24 + sigma_3(n/4)/3 - n*sigma(n/4)/4 + sigma(n/4)/24 + sigma_3(n/2)/16.
a(n) = (1/48)*(sigma_3(n) + 2*sigma(n) - 3*n*sigma(n)) + (1/768)*((1 + (-1)^n))*(173*sigma_3(n) - 21*sigma_3(2*n) + 28*sigma(n) - 12*sigma(2*n) - 168*n*sigma(n) + 72*n*sigma(2*n)). - Ridouane Oudra, Nov 23 2022

A374963 a(n) = Sum_{k=1..n-1} sigma(k)*sigma_3(n-k).

Original entry on oeis.org

0, 1, 12, 59, 200, 526, 1184, 2399, 4368, 7656, 12316, 19586, 29008, 43244, 60272, 85543, 114000, 156163, 200652, 266504, 333968, 432570, 528704, 673706, 806200, 1008644, 1192584, 1467684, 1707328, 2084676, 2390848, 2882487, 3286168, 3913722, 4409584, 5237489

Offset: 1

Chai Wah Wu, Jul 25 2024

Convolution of sigma with sigma_3.

In general, if k>=1, m>=1 and a(n) = Sum_{j=1..n-1} sigma_k(j) * sigma_m(n-j), then Sum_{j=1..n} a(j) ~ Gamma(k+1) * Gamma(m+1) * zeta(k+1) * zeta(m+1) * n^(k+m+2) / Gamma(k+m+3). - Vaclav Kotesovec, Sep 19 2024

Vaclav Kotesovec, Table of n, a(n) for n = 1..10000

Cf. A000203, A000385, A001158, A374974, A376290.

Table[Sum[DivisorSigma[1,k] *DivisorSigma[3,n-k],{k,n-1}],{n,36}] (* James C. McMahon, Aug 11 2024 *)

from sympy import divisor_sigma
def A374963(n): return sum(divisor_sigma(i)*divisor_sigma(n-i,3) for i in range(1,n))

Sum_{k=1..n} a(k) ~ Pi^6 * n^6 / 64800. - Vaclav Kotesovec, Sep 19 2024

A374978 a(n) = Sum_{i+j+k+l+m=n, i,j,k,l,m >= 1} sigma(i)sigma(j)sigma(k)sigma(l)sigma(m).

Original entry on oeis.org

0, 0, 0, 0, 1, 15, 110, 545, 2095, 6713, 18750, 47040, 108185, 231640, 467034, 894605, 1639680, 2891475, 4929660, 8155182, 13135080, 20651875, 31770970, 47923680, 70989801, 103454645, 148464520, 210155730, 293558265, 405325092, 553175000, 747508125, 999747750

Offset: 1

Chai Wah Wu, Jul 26 2024

nonn

5-fold convolution of A000203.

Convolution of A000203 and A374977.

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A000203, A000385, A319083, A374951, A374977.

b:= proc(n, k) option remember; `if`(k=0, `if`(n=0, 1, 0),
      `if`(k=1, `if`(n=0, 0, numtheory[sigma](n)), (q->
       add(b(j, q)*b(n-j, k-q), j=0..n))(iquo(k, 2))))
    end:
a:= n-> b(n, 5):
seq(a(n), n=1..55);  # Alois P. Heinz, Jul 26 2024

b[n_, k_] := b[n, k] = If[k == 0, If[n == 0, 1, 0], If[k == 1, If[n == 0, 0, DivisorSigma[1, n]], Function[q, Sum[b[j, q]*b[n - j, k - q], {j, 0, n}]][Quotient[k, 2]]]];
a[n_] := b[n, 5];
Table[a[n], {n, 1, 55}] (* Jean-François Alcover, Jul 11 2025, after Alois P. Heinz *)

from sympy import divisor_sigma
def A374978(n): return sum(divisor_sigma(j)*sum((5*divisor_sigma(i+1,3)-(5+6*i)*divisor_sigma(i+1))*(5*divisor_sigma(n-j-i-1,3)-(5+6*(n-j-i-2))*divisor_sigma(n-j-i-1)) for i in range(1,n-j-2)) for j in range(1,n))//144

a(n) = Sum_{i=1..n-1} A000203(i)*A374977(n-i).
a(n) = Sum_{i=1..n-2} A000385(i)*A374951(n-i-1).
Column k=5 of A319083.
Sum_{k=1..n} a(k) ~ Pi^10 * n^10 / 28217548800. - Vaclav Kotesovec, Sep 20 2024

A307306 Self-composition of the sum of divisors function (A000203).

Original entry on oeis.org

1, 6, 26, 101, 366, 1294, 4400, 14706, 48362, 157583, 507714, 1621211, 5138804, 16204008, 50867068, 159004142, 494928072, 1534638702, 4743180908, 14622202326, 44978845086, 138074363360, 422979847404, 1293101281551, 3945553307665, 12018461150832, 36556888102402

Offset: 1

Ilya Gutkovskiy, Apr 01 2019

nonn

Vaclav Kotesovec, Table of n, a(n) for n = 1..750
Eric Weisstein's World of Mathematics, Divisor Function

Cf. A000203, A000385, A051027, A307305.

g[x_] := g[x] = Sum[k x^k/(1 - x^k), {k, 1, 27}]; a[n_] := a[n] = SeriesCoefficient[g[g[x]], {x, 0, n}]; Table[a[n], {n, 27}]

G.f.: g(g(x)), where g(x) = Sum_{k>=1} k*x^k/(1 - x^k) is the g.f. of A000203.

A330088 a(n) = Sum_{k=1..n} binomial(n,k) * sigma(k) * sigma(n - k + 1), where sigma = A000203.

Original entry on oeis.org

1, 9, 43, 155, 511, 1442, 4131, 10323, 28171, 63987, 171667, 369395, 957958, 2047694, 5078963, 10671529, 26542339, 53522031, 132273403, 268623854, 647842889, 1266118858, 3197923083, 6058756355, 14581380971, 29480406552, 68634048862, 131847974143, 323289015466, 611887749996

Offset: 1

Ilya Gutkovskiy, Dec 03 2019

nonn

Cf. A000203, A000385, A007318, A185003, A328681.

[&+[Binomial(n,k)*DivisorSigma(1,k)*DivisorSigma(1,n-k+1):k in [1..n]]:n in [1..30]]; // Marius A. Burtea, Dec 03 2019

Table[Sum[Binomial[n, k] DivisorSigma[1, k] DivisorSigma[1, n - k + 1], {k, 1, n}], {n, 1, 30}]
nmax = 30; CoefficientList[Series[(1/2) D[Sum[DivisorSigma[1, k] x^k/k!, {k, 1, nmax}]^2, x], {x, 0, nmax}], x] Range[0, nmax]! // Rest

a(n) = sum(k=1, n, binomial(n,k)*sigma(k)*sigma(n-k+1)); \\ Michel Marcus, Dec 05 2019

E.g.f.: (1/2) * d/dx (Sum_{k>=1} sigma(k) * x^k / k!)^2.

A338865 Triangle T(n,k) defined by Sum_{k=1..n} T(n,k)u^kx^n/n! = Product_{j>0} ( exp(j*x^j/(1 - x^j)) )^u.

Original entry on oeis.org

1, 6, 1, 24, 18, 1, 168, 204, 36, 1, 720, 2280, 780, 60, 1, 8640, 25200, 14400, 2100, 90, 1, 40320, 292320, 252000, 58800, 4620, 126, 1, 604800, 3729600, 4334400, 1486800, 183120, 8904, 168, 1, 4717440, 46811520, 76265280, 35743680, 6335280, 474768, 15624, 216, 1

Offset: 1

Seiichi Manyama, Nov 13 2020

			exp(Sum_{n>0} u*sigma(n)*x^n) = 1 + u*x + (6*u+u^2)*x^2/2! + (24*u+18*u^2+u^3)*x^3/3! + ... .
Triangle begins:
       1;
       6,       1;
      24,      18,       1;
     168,     204,      36,       1;
     720,    2280,     780,      60,      1;
    8640,   25200,   14400,    2100,     90,    1;
   40320,  292320,  252000,   58800,   4620,  126,   1;
  604800, 3729600, 4334400, 1486800, 183120, 8904, 168, 1;
  ...

Peter Luschny, The Bell transform.

Column k=1..2 give n! * sigma(n), (n!/2) * A000385(n-1).
Rows sum give A294361.
Cf. A000203 (sigma(n)), A008298, A338864, A338871.

T[n_, 0] := Boole[n == 0]; T[n_, k_] := T[n, k] = Sum[Boole[j > 0] * Binomial[n - 1, j - 1] * j! * DivisorSigma[1, j] * T[n - j, k - 1], {j, 0, n - k + 1}]; Table[T[n, k], {n, 1, 9}, {k, 1, n}] // Flatten (* Amiram Eldar, Nov 13 2020 *)

{T(n, k) = my(u='u); n!*polcoef(polcoef(prod(j=1, n, exp(j*x^j/(1-x^j+x*O(x^n)))^u), n), k)}

a(n) = if(n<1, 0, n!*sigma(n));
T(n, k) = if(k==0, 0^n, sum(j=0, n-k+1, binomial(n-1, j-1)*a(j)*T(n-j, k-1)))

E.g.f.: exp(Sum_{n>0} u*sigma(n)*x^n).
T(n; u) = Sum_{k=1..n} T(n,k)*u^k is given by T(n; u) = u * (n-1)! * Sum_{k=1..n} k*sigma(k)*T(n-k; u)/(n-k)!, T(0; u) = 1.
T(n,k) = (n!/k!) * Sum_{i_1,i_2,...,i_k > 0 and i_1+i_2+...+i_k=n} Product_{j=1..k} sigma(i_j).

A374979 a(n) = Sum_{i+j+k+l+m+r=n, i,j,k,l,m,r >= 1} sigma(i)sigma(j)sigma(k)sigma(l)sigma(m)*sigma(r).

Original entry on oeis.org

0, 0, 0, 0, 0, 1, 18, 159, 942, 4281, 16050, 51932, 149532, 391524, 947246, 2143677, 4581204, 9316195, 18138636, 33984912, 61534652, 108055425, 184582014, 307515038, 500798058, 798762453, 1249917936, 1921788036, 2907159804, 4332046200, 6365441400, 9232216725

Offset: 1

Chai Wah Wu, Jul 26 2024

nonn

6-fold convolution of A000203.
Convolution of A000203 and A374978.
a(n) = Sum_{i=1..n-1} A000203(i)*A374978(n-i).
a(n) = Sum_{i=1..n-2} A000385(i)*A374977(n-i-1).
a(n) = Sum_{i=1..n-1} A374951(i)*A374951(n-i).
a(n) = Sum_{i+j+k=n-3, i,j,k>=1} A000385(i)*A000385(j)*A000385(k).
Column k=6 of A319083.
In general, if the sequence "a" is a k-fold convolution of A000203, then Sum_{k=1..n} a(k) ~ Pi^(2*k) * n^(2*k) / (6^k * (2*k)!). - Vaclav Kotesovec, Sep 20 2024

Alois P. Heinz, Table of n, a(n) for n = 1..10000

Cf. A000203, A000385, A319083, A374951, A374977, A374978.

b:= proc(n, k) option remember; `if`(k=0, `if`(n=0, 1, 0),
      `if`(k=1, `if`(n=0, 0, numtheory[sigma](n)), (q->
       add(b(j, q)*b(n-j, k-q), j=0..n))(iquo(k, 2))))
    end:
a:= n-> b(n, 6):
seq(a(n), n=1..55);  # Alois P. Heinz, Jul 26 2024

from functools import lru_cache
from sympy import divisor_sigma
def A374979(n):
    @lru_cache(maxsize=None)
    def g(x):
        f = factorint(x+1).items()
        return(5*prod((p**(3*(e+1))-1)//(p**3-1) for p,e in f)-(5+6*x)*prod((p**(e+1)-1)//(p-1) for p, e in f))//12
    return sum(g(i)*g(j)*g(n-3-i-j) for i in range(1,n-4) for j in range(1,n-i-3))

Sum_{k=1..n} a(k) ~ Pi^12 * n^12 / 22348298649600. - Vaclav Kotesovec, Sep 20 2024

A341637 a(n) = Sum_{d|n} phi(d) * sigma(d) * sigma(n/d).

Original entry on oeis.org

1, 6, 12, 30, 30, 72, 56, 138, 123, 180, 132, 360, 182, 336, 360, 602, 306, 738, 380, 900, 672, 792, 552, 1656, 795, 1092, 1176, 1680, 870, 2160, 992, 2538, 1584, 1836, 1680, 3690, 1406, 2280, 2184, 4140, 1722, 4032, 1892, 3960, 3690, 3312, 2256, 7224, 2835, 4770, 3672, 5460

Offset: 1

Ilya Gutkovskiy, Feb 16 2021

Michael De Vlieger, Table of n, a(n) for n = 1..10000

Cf. A000010, A000203, A000385, A034761, A038040, A062354, A062952, A183699, A330523.

Table[Sum[EulerPhi[d] DivisorSigma[1, d] DivisorSigma[1, n/d], {d, Divisors[n]}], {n, 52}]
Table[Sum[DivisorSigma[1, GCD[n, k]] DivisorSigma[1, n/GCD[n, k]], {k, n}], {n, 52}]
f[p_, e_] := (p^(2*e + 3) - (e + 1)*(p^2 - 1)*p^e - p)/((p - 1)^2*(p + 1)); a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 50] (* Amiram Eldar, Nov 12 2022 *)

a(n) = sumdiv(n, d, eulerphi(d)*sigma(d)*sigma(n/d)); \\ Michel Marcus, Feb 17 2021

a(n) = Sum_{k=1..n} sigma(gcd(n,k)) * sigma(n/gcd(n,k)).
From Amiram Eldar, Nov 12 2022: (Start)
Multiplicative with a(p^e) = (p^(2*e+3) - (e+1)*(p^2-1)*p^e - p)/((p-1)^2*(p+1)).
Sum_{k=1..n} a(k) ~ c * n^3, where c = (zeta(2)*zeta(3)/3) * Product_{p prime} (1 - 1/(p^2*(p+1))) = (1/3) * A183699 * A330523 = 0.581007... . (End)

A346193 Convolution of level 5 of the divisor function.

Original entry on oeis.org

0, 0, 0, 0, 0, 1, 3, 4, 7, 6, 15, 17, 27, 34, 36, 52, 64, 75, 91, 102, 122, 155, 169, 193, 228, 263, 276, 326, 349, 415, 430, 500, 520, 620, 681, 727, 741, 881, 880, 1090, 1020, 1192, 1178, 1375, 1513, 1590, 1557, 1809, 1756, 2274, 2024, 2323, 2245, 2626, 2865

Offset: 1

Amiram Eldar, Jul 09 2021

nonn

Amiram Eldar, Table of n, a(n) for n = 1..10000
Şaban Alaca and Kenneth S. Williams, Evaluation of the convolution sums ..., Journal of Number Theory, Vol. 124, No. 2 (2007) pp. 491-510.
Mathieu Lemire and Kenneth S. Williams, Evaluation of two convolution sums involving the sum of divisors function, Bulletin of the Australian Mathematical Society, Vol. 73, No. 1 (2006), pp. 107-115.
Emmanuel Royer, Evaluating convolution sums of the divisor function by quasimodular forms, International Journal of Number Theory, Vol. 3, No. 2 (2007) pp. 231-261.

Cf. A000203, A000385, A001158, A030210, A218276, A218277, A218278.

a[n_] := Sum[DivisorSigma[1, k] * DivisorSigma[1, n - 5*k], {k, 1, (n - 1)/5}]; Array[a, 100]
(* or *)
c[n_] := SeriesCoefficient[q * (QPochhammer[q] * QPochhammer[q^5])^4, {q, 0, n}]; a[n_] := 5 * DivisorSigma[3, n]/312 + If[Divisible[n, 5], 125 * DivisorSigma[3, n/5]/312, 0] - n * DivisorSigma[1, n]/20 - If[Divisible[n, 5], n * DivisorSigma[1, n/5]/4, 0] + DivisorSigma[1, n]/24 + If[Divisible[n, 5], DivisorSigma[1, n/5]/24, 0] - c[n]/130; Array[a, 100]

a(n) = Sum_{k < n/5} sigma(k) * sigma(n-5*k).

a(n) = 5*sigma_3(n)/312 + 125*sigma_3(n/5)/312 + (1/24 - n/20)*sigma(n) + (1/24 - n/4)*sigma(n/5) - c_5(n)/130, where sigma_3(n/5) = sigma(n/5) = 0 if n is not divisible by 5, and c_5(n) is the coefficient of q^n in the expansion of (eta(q) * eta(q^5))^4 (A030210).

A379635 Triangle read by rows: T(n,k) = A000203(k)*A000203(n-k+1), n >= 1, k >= 1.

Original entry on oeis.org

1, 3, 3, 4, 9, 4, 7, 12, 12, 7, 6, 21, 16, 21, 6, 12, 18, 28, 28, 18, 12, 8, 36, 24, 49, 24, 36, 8, 15, 24, 48, 42, 42, 48, 24, 15, 13, 45, 32, 84, 36, 84, 32, 45, 13, 18, 39, 60, 56, 72, 72, 56, 60, 39, 18, 12, 54, 52, 105, 48, 144, 48, 105, 52, 54, 12, 28, 36, 72, 91, 90, 96, 96, 90, 91, 72, 36, 28

Offset: 1

Omar E. Pol, Jan 14 2025

			Triangle begins:
   1;
   3,   3;
   4,   9,   4;
   7,  12,  12,   7;
   6,  21,  16,  21,   6;
  12,  18,  28,  28,  18,  12;
   8,  36,  24,  49,  24,  36,   8;
  15,  24,  48,  42,  42,  48,  24,  15;
  13,  45,  32,  84,  36,  84,  32,  45,  13;
  18,  39,  60,  56,  72,  72,  56,  60,  39,  18;
  12,  54,  52, 105,  48, 144,  48, 105,  52,  54,  12;
  28,  36,  72,  91,  90,  96,  96,  90,  91,  72,  36,  28;
  14,  84,  48, 126,  78, 180,  64, 180,  78, 126,  48,  84,  14;
  ...
For n = 10 the calculation of the row 10 is as follows:
    k    A000203         T(10,k)
    1       1   *  18   =   18
    2       3   *  13   =   39
    3       4   *  15   =   60
    4       7   *   8   =   56
    5       6   *  12   =   72
    6      12   *   6   =   72
    7       8   *   7   =   56
    8      15   *   4   =   60
    9      13   *   3   =   39
   10      18   *   1   =   18
                 A000203
.

Index entries for sequences related to sigma(n).

Column 1 and leading diagonal give A000203.
Middle diagonal gives A072861.
Row sums give A000385.
Cf. A221529.

T[n_,k_]:=DivisorSigma[1,k]*DivisorSigma[1,n-k+1];Table[T[n,k],{n,12},{k,n }]//Flatten (* James C. McMahon, Jan 15 2025 *)

PARI
```
T(n, k)=sigma(k)*sigma(n-k+1)
```

cp's OEIS Frontend

A218278 Convolution of level 4 of the divisor function.

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A374963 a(n) = Sum_{k=1..n-1} sigma(k)*sigma_3(n-k).

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A374978 a(n) = Sum_{i+j+k+l+m=n, i,j,k,l,m >= 1} sigma(i)*sigma(j)*sigma(k)*sigma(l)*sigma(m).

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A307306 Self-composition of the sum of divisors function (A000203).

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A330088 a(n) = Sum_{k=1..n} binomial(n,k) * sigma(k) * sigma(n - k + 1), where sigma = A000203.

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A338865 Triangle T(n,k) defined by Sum_{k=1..n} T(n,k)*u^k*x^n/n! = Product_{j>0} ( exp(j*x^j/(1 - x^j)) )^u.

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A374979 a(n) = Sum_{i+j+k+l+m+r=n, i,j,k,l,m,r >= 1} sigma(i)*sigma(j)*sigma(k)*sigma(l)*sigma(m)*sigma(r).

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A341637 a(n) = Sum_{d|n} phi(d) * sigma(d) * sigma(n/d).

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A374978 a(n) = Sum_{i+j+k+l+m=n, i,j,k,l,m >= 1} sigma(i)sigma(j)sigma(k)sigma(l)sigma(m).

A338865 Triangle T(n,k) defined by Sum_{k=1..n} T(n,k)u^kx^n/n! = Product_{j>0} ( exp(j*x^j/(1 - x^j)) )^u.

A374979 a(n) = Sum_{i+j+k+l+m+r=n, i,j,k,l,m,r >= 1} sigma(i)sigma(j)sigma(k)sigma(l)sigma(m)*sigma(r).