A319083 -id:A319083 - cp's OEIS frontend

A000385 Convolution of A000203 with itself.

1, 6, 17, 38, 70, 116, 185, 258, 384, 490, 686, 826, 1124, 1292, 1705, 1896, 2491, 2670, 3416, 3680, 4602, 4796, 6110, 6178, 7700, 7980, 9684, 9730, 12156, 11920, 14601, 14752, 17514, 17224, 21395, 20406, 24590, 24556, 28920, 27860, 34112, 32186, 38674, 37994, 43980, 42136, 51646, 47772, 56749, 55500, 64316, 60606, 73420, 67956, 80500, 77760, 88860, 83810, 102284, 92690, 108752, 105236, 120777, 112672, 135120, 123046, 145194, 138656, 157512, 146580, 177515, 159396, 185744, 179122

Offset: 1

N. J. A. Sloane

Convolution of A340793 and A024916. - Omar E. Pol, Feb 17 2021

N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
Jacques Touchard, On prime numbers and perfect numbers, Scripta Math., 129 (1953), 35-39.

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
Nicolae Ciprian Bonciocat, Congruences for the Convolution of Divisor sum function, Bull. Greek Math. Soc., Vol 47 (2003), pp. 19-29.
MathOverflow, Searching for a proof for a series identity.
Srinivasa Ramanujan, On certain arithmetical functions, Transactions of the Cambridge Philosophical Society, 22, No.9 (1916), 169- 184 (see Table IV, line 1).
Jacques Touchard, On prime numbers and perfect numbers, Scripta Math., 129 (1953), 35-39. [Annotated scanned copy]

Cf. A000203, A024916, A001158, A340793.

Column k=2 of A319083 (shifted).

a000385 n = sum $ zipWith (*) sigmas $ reverse sigmas where
   sigmas = take n a000203_list
-- Reinhard Zumkeller, Sep 20 2011

f:= n -> 5/12*numtheory:-sigma[3](n+1)-(5+6*n)/12*numtheory:-sigma(n+1):
map(f, [$1..100]); # Robert Israel, Sep 17 2018

a[n_] := Sum[DivisorSigma[1, k] DivisorSigma[1, n-k+1], {k, 1, n}];
Array[a, 100] (* Jean-François Alcover, Aug 01 2018 *)

a(n) = sum(k=1, n, sigma(k)*sigma(n-k+1)); \\ Michel Marcus, Nov 10 2016

a(n) = my(f = factor(n+1)); (5 * sigma(f, 3) - (6*n + 5) * sigma(f)) / 12; \\ Amiram Eldar, Jan 04 2025

from sympy import factorint
def A000385(n):
    f = factorint(n+1).items()
    return(5*prod((p**(3*(e+1))-1)//(p**3-1) for p,e in f)-(5+6*n)*prod((p**(e+1)-1)//(p-1) for p, e in f))//12 # Chai Wah Wu, Jul 25 2024

a(n) = Sum_{k=1..n} A000203(k)*A000203(n-k+1).
G.f.: (1/x)*(Sum_{k>=1} k*x^k/(1 - x^k))^2. - Ilya Gutkovskiy, Nov 10 2016
a(5*n+1)==0 (mod 5) and a(7*n+6)==0 (mod 7). See Bonciocat link. - Michel Marcus, Nov 10 2016
a(n) = (5/12)*A001158(n+1) - ((5+6*n)/12)*A000203(n+1). - Robert Israel, Sep 17 2018
Sum_{k=1..n} a(k) ~ Pi^4 * n^4 / 864. - Vaclav Kotesovec, Apr 02 2019

More terms from Sean A. Irvine, Nov 14 2010

A180305 G.f.: 1/(1 + x*d/dx log(eta(x))), where eta(x) is Dedekind's eta(q) function without the q^(1/24) factor.

Original entry on oeis.org

1, 1, 4, 11, 34, 96, 288, 833, 2456, 7175, 21054, 61633, 180674, 529220, 1550800, 4543446, 13312552, 39004278, 114281748, 334837511, 981059294, 2874447292, 8421986238, 24675963950, 72299290794, 211833080161, 620659794584, 1818500391218, 5328110328116, 15611082044176, 45739647180588, 134014753120706, 392656158141832

Offset: 0

Paul D. Hanna, Jan 18 2011

nonn

INVERT transform of sigma (A000203). - Alois P. Heinz, Feb 11 2021

			G.f.: A(x) = 1 + x + 4*x^2 + 11*x^3 + 34*x^4 + 96*x^5 + 288*x^6 +...
eta(x)^3/A(x) = 1 - 4*x + 10*x^3 - 21*x^6 + 39*x^10 - 66*x^15 + 104*x^21 +...+ A184363(n)*x^n +...
1 + x*d/dx log(eta(x)) = 1 - x - 3*x^2 - 4*x^3 - 7*x^4 - 6*x^5 - 12*x^6 - 8*x^7 - 15*x^8 +...+ -sigma(n)*x^n +...

Vaclav Kotesovec, Table of n, a(n) for n = 0..1000

Cf. A184363, A000203, A000041.

Row sums of A319083.

a:= proc(n) option remember; `if`(n=0, 1,
      add(a(n-i)*numtheory[sigma](i), i=1..n))
    end:
seq(a(n), n=0..36);  # Alois P. Heinz, Feb 11 2021

nmax = 50; CoefficientList[Series[1/(1 - Sum[DivisorSigma[1, k]*x^k, {k, 1, nmax}]), {x, 0, nmax}], x] (* Vaclav Kotesovec, Jan 06 2017 *)

{a(n)=polcoeff(1/(1+x*deriv(log(eta(x+x*O(x^n))))), n)}

{a(n)=if(n==0,1,sum(k=0,n-1,sigma(n-k)*a(k)))}

N=66; x='x+O('x^N); Vec(1/(1 - sum(k=1,N, x^k/(1-x^k)^2))) \\ Joerg Arndt, Mar 09 2014

a(n) = Sum_{k=0,n-1} sigma(n-k)*a(k) for n>0 with a(0) = 1.
G.f.: 1/(1 - sum(k>=1, x^k/(1-x^k)^2)). [Joerg Arndt, Mar 09 2014]
a(n) ~ c * d^n, where d = 2.92994725111235280869138453465150817383965264075630759525007993985560038385... is the root of the equation Sum_{k>=1} sigma(k)/d^k = 1 and c = 0.45133473613134383104139698267531812019856702278773719486399141396046228911... - Vaclav Kotesovec, Jul 28 2018

A374951 a(n) = Sum_{i+j+k=n, i,j,k >= 1} sigma(i) * sigma(j) * sigma(k).

Original entry on oeis.org

0, 0, 1, 9, 39, 120, 300, 645, 1261, 2262, 3825, 6160, 9471, 14178, 20376, 28965, 39600, 54066, 71145, 94248, 120140, 155310, 193116, 244560, 297819, 370860, 443710, 544554, 641655, 778458, 904800, 1085445, 1248762, 1483308, 1688052, 1991515, 2244375, 2626380

Offset: 1

Seiichi Manyama, Jul 25 2024

nonn

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

Column k=3 of A319083.

Cf. A000203, A000385, A191829.

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, 3):
seq(a(n), n=1..55);  # Alois P. Heinz, Jul 25 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, 3];
Table[a[n], {n, 1, 55}] (* Jean-François Alcover, Mar 14 2025, after Alois P. Heinz *)

my(N=40, x='x+O('x^N)); concat([0, 0], Vec(sum(k=1, N, k*x^k/(1-x^k))^3))

from sympy import divisor_sigma
def A374951(n): return (60*sum(divisor_sigma(i)*divisor_sigma(n-i,3) for i in range(1,n))+divisor_sigma(n)*(9*n*(2*n-1)+1)-5*divisor_sigma(n,3)*(3*n-1))//144  # Chai Wah Wu, Jul 25 2024

G.f.: ( Sum_{k>=1} k * x^k/(1 - x^k) )^3 = ( Sum_{k>=1} x^k/(1 - x^k)^2 )^3.
a(n) = Sum_{i=1..n-2} sigma(i)*A000385(n-i-1). - Chai Wah Wu, Jul 25 2024
Sum_{k=1..n} a(k) ~ Pi^6 * n^6 / 155520. - Vaclav Kotesovec, Sep 19 2024

A374977 a(n) = Sum_{i+j+k+l=n, i,j,k,l >= 1} sigma(i)sigma(j)sigma(k)*sigma(l).

Original entry on oeis.org

0, 0, 0, 1, 12, 70, 280, 885, 2364, 5586, 12000, 23870, 44660, 79272, 134768, 220565, 349440, 538270, 807840, 1187004, 1706840, 2415150, 3354120, 4601870, 6209612, 8303610, 10935960, 14309640, 18460260, 23708184, 30044000, 37967925, 47368480, 59022432, 72633816

Offset: 1

Chai Wah Wu, Jul 26 2024

nonn

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

Cf. A000203, A000385, A319083, A374951.

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, 4];
Table[a[n], {n, 1, 35}] (* Jean-François Alcover, Jul 11 2025, after Alois P. Heinz in A319083 *)

from sympy import divisor_sigma
def A374977(n): return sum((5*divisor_sigma(i+1,3)-(5+6*i)*divisor_sigma(i+1))*(5*divisor_sigma(n-i-1,3)-(5+6*(n-i-2))*divisor_sigma(n-i-1)) for i in range(1,n-2))//144

4-fold convolution of A000203.
Convolution of A000203 and A374951.
Convolution of A000385 with itself.
a(n) = Sum_{i=1..n-1} A000203(i)*A374951(n-i).
a(n) = Sum_{i=1..n-3} A000385(i)*A000385(n-i-2).
Column k=4 of A319083.
Sum_{k=1..n} a(k) ~ Pi^8 * n^8 / 52254720. - Vaclav Kotesovec, Sep 20 2024

A283334 G.f.: 1/(1 - x*d/dx log(eta(x))), where eta(x) is Dedekind's eta(q) function without the q^(1/24) factor.

Original entry on oeis.org

1, -1, -2, 1, 2, 4, -6, -5, 4, 1, 18, -13, -26, 4, 22, 66, -76, -78, 66, 37, 122, -136, -144, 10, 54, 599, -368, -746, 196, 568, 744, -938, -156, -312, -1428, 2720, 3340, -2324, -8588, 1520, 8814, 4846, 1380, -16565, -16966, -6324, 79170, 47250, -160346

Offset: 0

Seiichi Manyama, Mar 05 2017

sign

			G.f.: A(x) = 1 - x - 2*x^2 + x^3 + 2*x^4 + 4*x^5 - 6*x^6 - 5*x^7 + ...
1/A(x) = 1 - x*d/dx log(eta(x)) = 1 + x + 3*x^2 + 4*x^3 + 7*x^4 + ... + sigma(n)*x^n + ...
eta(x)^3/A(x) = 1 - 2*x + 7*x^6 - 21*x^10 + 44*x^15 - 78*x^21 + ... + A184366(n)*x^n + ...

Seiichi Manyama, Table of n, a(n) for n = 0..1000
Bernhard Heim, Markus Neuhauser, and Robert Troeger, Zeros Transfer For Recursively defined Polynomials, arXiv:2304.02694 [math.NT], 2023. See Table 5 p. 14.

Cf. A180305 (1/(1 + x*d/dx log(eta(x)))), A184366, A319083.

lista(nn) = {va = vector(nn); va[1] = 1; for (n=1, nn-1, va[n+1] = -sum(k=0, n-1, sigma(n-k)*va[k+1]);); va;} \\ Michel Marcus, Mar 05 2017

a(n) = -Sum_{k=0..n-1} sigma(n-k)*a(k) for n>0 with a(0) = 1.
G.f.: 1/(1 + Sum_{k>=1} k*x^k/(1 - x^k)). - Ilya Gutkovskiy, Oct 18 2018
a(n) = Sum_{k=0..n} (-1)^k * A319083(n,k). - Alois P. Heinz, Feb 07 2025

A320019 Coefficients of polynomials related to the number of divisors, triangle read by rows, T(n,k) for 0 <= k <= n.

Original entry on oeis.org

1, 0, 1, 0, 2, 1, 0, 2, 4, 1, 0, 3, 8, 6, 1, 0, 2, 14, 18, 8, 1, 0, 4, 20, 41, 32, 10, 1, 0, 2, 28, 78, 92, 50, 12, 1, 0, 4, 37, 132, 216, 175, 72, 14, 1, 0, 3, 44, 209, 440, 490, 298, 98, 16, 1, 0, 4, 58, 306, 814, 1172, 972, 469, 128, 18, 1

Offset: 0

Peter Luschny, Oct 03 2018

Column k is the k-fold self-convolution of tau (A000005). - Alois P. Heinz, Feb 01 2021

			Triangle starts:
[0] 1
[1] 0, 1
[2] 0, 2,  1
[3] 0, 2,  4,   1
[4] 0, 3,  8,   6,   1
[5] 0, 2, 14,  18,   8,   1
[6] 0, 4, 20,  41,  32,  10,   1
[7] 0, 2, 28,  78,  92,  50,  12,  1
[8] 0, 4, 37, 132, 216, 175,  72, 14,  1
[9] 0, 3, 44, 209, 440, 490, 298, 98, 16, 1

Alois P. Heinz, Rows n = 0..200, flattened

Columns k=0-4 give: A000007, A000005, A055507, A191829, A375002.
Row sums are A129921.
T(2n,n) gives A340992.
Cf. A319083.

P := proc(n, x) option remember; if n = 0 then 1 else
x*add(numtheory:-tau(n-k)*P(k,x), k=0..n-1) fi end:
Trow := n -> seq(coeff(P(n, x), x, k), k=0..n):
seq(lprint([n], Trow(n)), n=0..9);
# second Maple program:
T:= proc(n, k) option remember; `if`(k=0, `if`(n=0, 1, 0),
      `if`(k=1, `if`(n=0, 0, numtheory[tau](n)), (q->
       add(T(j, q)*T(n-j, k-q), j=0..n))(iquo(k, 2))))
    end:
seq(seq(T(n, k), k=0..n), n=0..12);  # Alois P. Heinz, Feb 01 2021
# Uses function PMatrix from A357368.
PMatrix(10, NumberTheory:-tau); # Peter Luschny, Oct 19 2022

T[n_, k_] := T[n, k] = If[k == 0, If[n == 0, 1, 0],
     If[k == 1, If[n == 0, 0, DivisorSigma[0, n]],
     With[{q = Quotient[k, 2]}, Sum[T[j, q]*T[n-j, k-q], {j, 0, n}]]]];
Table[Table[T[n, k], {k, 0, n}], {n, 0, 12}] // Flatten (* Jean-François Alcover, Feb 11 2021, after Alois P. Heinz *)

The polynomials are defined by recurrence: p(0,x) = 1 and for n > 0 by
p(n, x) = x*Sum_{k=0..n-1} tau(n-k)*p(k, x).
Sigma[k](n) computes the sum of the k-th power of positive divisors of n. The recurrence applied with k = 0 gives this triangle, with k = 1 gives A319083.
T(n,k) = [x^n] (Sum_{j>=1} tau(j)*x^j)^k. - Alois P. Heinz, Feb 14 2021

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

A340993 a(n) is the (2n)-th term of the n-fold self-convolution of the sum of divisors function sigma.

Original entry on oeis.org

1, 3, 17, 120, 885, 6713, 51932, 407214, 3224845, 25733325, 206584437, 1666561042, 13498994796, 109713432390, 894291885000, 7307812510970, 59847327807597, 491062976039618, 4036174402666925, 33224883837921930, 273873806179142545, 2260338391869532332

Offset: 0

Alois P. Heinz, Feb 01 2021

nonn

Alois P. Heinz, Table of n, a(n) for n = 0..1080

Cf. A000203, A319083.

b:= proc(n, k) option remember; `if`(k=0, 1,
      `if`(k=1, numtheory[sigma](n+1), (q->
       add(b(j, q)*b(n-j, k-q), j=0..n))(iquo(k, 2))))
    end:
a:= n-> b(n$2):
seq(a(n), n=0..23);

a(n) = [x^(2n)] (Sum_{j>=1} sigma(j)*x^j)^n.
a(n) = A319083(2n,n).
a(n) ~ c * d^n / sqrt(n), where d = 8.455610430383829836198938524980234226695900064615457328971640722426861925... and c = 0.352126317954512592610958969393229871240824031029408304023118123356... - Vaclav Kotesovec, Jul 25 2024

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

cp's OEIS Frontend

A000385 Convolution of A000203 with itself.

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A180305 G.f.: 1/(1 + x*d/dx log(eta(x))), where eta(x) is Dedekind's eta(q) function without the q^(1/24) factor.

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A374951 a(n) = Sum_{i+j+k=n, i,j,k >= 1} sigma(i) * sigma(j) * sigma(k).

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A374977 a(n) = Sum_{i+j+k+l=n, i,j,k,l >= 1} sigma(i)*sigma(j)*sigma(k)*sigma(l).

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A283334 G.f.: 1/(1 - x*d/dx log(eta(x))), where eta(x) is Dedekind's eta(q) function without the q^(1/24) factor.

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A320019 Coefficients of polynomials related to the number of divisors, triangle read by rows, T(n,k) for 0 <= k <= n.

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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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A374977 a(n) = Sum_{i+j+k+l=n, i,j,k,l >= 1} sigma(i)sigma(j)sigma(k)*sigma(l).

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).

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).