A000385 -id:A000385 - cp's OEIS frontend

A259694 a(n) = Sum(k^6sigma(k)sigma(n-k),k=1..n-1).

0, 1, 195, 3496, 38195, 192780, 977386, 3216320, 9860049, 27321870, 65803045, 144005856, 308944925, 635774072, 1112550390, 2153146880, 3618341556, 6391671525, 9949570455, 16725562160, 24691972080, 40979569092, 56807498030, 89450231040, 120404165825

Offset: 1

N. J. A. Sloane, Jul 03 2015

nonn

This was formerly A001479.

Colin Barker, Table of n, a(n) for n = 1..1000
J. Touchard, On prime numbers and perfect numbers, Scripta Math., 129 (1953), 35-39. [Annotated scanned copy]

Cf. A000385, A000441, A000477, A000499, A259692, A259693, A259695, A259696.

S:=(n,e)->add(k^e*sigma(k)*sigma(n-k),k=1..n-1); f:=e->[seq(S(n,e),n=1..30)]; f(6);

a(n) = sum(k=1, n-1, k^6*sigma(k)*sigma(n-k)) \\ Colin Barker, Jul 16 2015

A259695 a(n) = Sum_{k=1..n-1} k^7 * sigma(k) * sigma(n-k).

Original entry on oeis.org

0, 1, 387, 9904, 142475, 850500, 5287786, 19400960, 68736681, 210682950, 565317445, 1328193216, 3163440917, 6945663368, 13045807350, 26914795520, 48673795956, 89900901837, 149363037975, 262436871200, 409003474320, 711715515852, 1035199173422, 1683466675200

Offset: 1

N. J. A. Sloane, Jul 03 2015

nonn

This was formerly A001480.

Colin Barker, Table of n, a(n) for n = 1..1000
J. Touchard, On prime numbers and perfect numbers, Scripta Math., 129 (1953), 35-39. [Annotated scanned copy]

Cf. A000203, A000385, A000441, A000477, A000499, A259692, A259693, A259694, A259696.

S:=(n,e)->add(k^e*sigma(k)*sigma(n-k),k=1..n-1); f:=e->[seq(S(n,e),n=1..30)]; f(7);

Table[Sum[k^7 DivisorSigma[1,k]DivisorSigma[1,n-k],{k,n-1}],{n,30}] (* Harvey P. Dale, Dec 14 2015 *)

a(n) = sum(k=1, n-1, k^7*sigma(k)*sigma(n-k)) \\ Colin Barker, Jul 16 2015

A259696 a(n) = Sum_{k=1..n-1} k^8sigma(k)sigma(n-k).

Original entry on oeis.org

0, 1, 771, 28552, 540563, 3830364, 29209978, 119337536, 490114881, 1659932478, 4961414965, 12516905184, 33139873949, 77515802840, 156374512326, 344012784128, 669604434612, 1292506329141, 2292202227639, 4210108803824, 6929184038448, 12639642518772, 19287324979742, 32384260599552

Offset: 1

N. J. A. Sloane, Jul 03 2015

nonn

This was formerly A001481.

Colin Barker, Table of n, a(n) for n = 1..1000
J. Touchard, On prime numbers and perfect numbers, Scripta Math., 129 (1953), 35-39. [Annotated scanned copy]

Cf. A000385, A000441, A000477, A000499, A259692, A259693, A259694, A259695.

S:=(n,e)->add(k^e*sigma(k)*sigma(n-k),k=1..n-1); f:=e->[seq(S(n,e),n=1..30)]; f(8);

a(n) = sum(k=1, n-1, k^8*sigma(k)*sigma(n-k)) \\ Colin Barker, Jul 16 2015

A065093 Convolution of A000010 with itself.

Original entry on oeis.org

1, 2, 5, 8, 16, 20, 36, 44, 68, 76, 120, 124, 188, 196, 276, 272, 404, 380, 544, 532, 716, 668, 968, 860, 1184, 1120, 1472, 1332, 1896, 1624, 2204, 2036, 2656, 2352, 3284, 2752, 3684, 3356, 4324, 3744, 5192, 4312, 5720, 5180, 6540, 5628, 7768, 6388, 8476

Offset: 1

Vladeta Jovovic, Nov 11 2001

Vaclav Kotesovec, Table of n, a(n) for n = 1..10000 (terms 1..1000 from Harry J. Smith)
A. E. Ingham, Some asymptotic formulae in the theory of numbers, Journal of the London Mathematical Society, Vol. s1-2, No. 3 (1927), pp. 202-208.

Cf. A000010, A000385, A055507, A065474, A330319.

Column k=2 of A340995.

Table[Sum[EulerPhi[j]*EulerPhi[n-j], {j, 1, n-1}], {n, 2, 50}] (* Vaclav Kotesovec, Aug 18 2021 *)

{ for (n=1, 1000, a=sum(k=1, n, eulerphi(k)*eulerphi(n+1-k)); write("b065093.txt", n, " ", a) ) } \\ Harry J. Smith, Oct 06 2009

a(n) = Sum_{k=1..n} phi(k)*phi(n+1-k), where phi is Euler totient function (A000010).
G.f.: (1/x)*(Sum_{k>=1} mu(k)*x^k/(1 - x^k)^2)^2. - Ilya Gutkovskiy, Jan 31 2017
a(n) ~ (n^3/6) * c * Product_{primes p|n+1} ((p^3-2*p+1)/(p*(p^2-2))), where c = Product_{p prime} (1 - 2/p^2) = 0.322634... (A065474) (Ingham, 1927). - Amiram Eldar, Jul 13 2024

A112964 Sum(mu(i)*sigma(j): i+j=n), with mu=A008683 and sigma=A000203.

Original entry on oeis.org

0, 1, 2, 0, 0, -6, -3, -12, -11, -13, -22, -19, -20, -30, -41, -15, -55, -24, -52, -41, -59, -24, -109, -22, -78, -42, -111, -26, -131, -2, -119, -75, -133, -8, -214, 7, -175, -68, -176, -17, -209, 14, -231, -73, -175, 45, -349, -11, -236, -20, -236, -53, -384, 68, -321, -56, -270, 1, -457, 41, -328, -48

Offset: 1

Reinhard Zumkeller, Oct 07 2005

sign

			a(5)=mu(1)*sigma(4)+mu(2)*sigma(3)+mu(3)*sigma(2)+mu(4)*sigma(1)
= 1*7 - 1*4 - 1*3 + 0*1 = 0.

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000

Cf. A024916, A000385, A068341, A112962, A112963, A112966, A112968.

a(n)=sum(i=1,n-1,moebius(i)*sigma(n-i)) \\ Charles R Greathouse IV, Feb 14 2013

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

A059356 A diagonal of triangle in A008298.

Original entry on oeis.org

1, 9, 59, 450, 3394, 30912, 293292, 3032208, 36290736, 433762560, 5925016800, 83648747520, 1335385128960, 20323375994880, 376785057196800, 6493118120294400, 132672192555571200, 2513351450024755200, 56577426980420505600, 1188283280226545664000, 29682641812682686464000, 658094690655791972352000

Offset: 2

N. J. A. Sloane, Jan 27 2001

nonn

L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 159.

Seiichi Manyama, Table of n, a(n) for n = 2..448

Cf. A000203, A000385, A008298.

nmax = 30; Table[n!/2 * Sum[DivisorSigma[1, k] * DivisorSigma[1, n-k] / k / (n-k), {k, 1, n-1}], {n, 2, nmax}] (* Vaclav Kotesovec, Nov 09 2020 *)

{a(n) = my(t='t); n!*polcoef(polcoef(prod(k=1, n, (1-x^k+x*O(x^n))^(-t)), n), 2)} \\ Seiichi Manyama, Nov 07 2020

{a(n)= (n-1)!*sum(k=1, n-1, sigma(k)*sigma(n-k)/k)} \\ Seiichi Manyama, Nov 09 2020

{a(n)= n!*sum(k=1, n-1, sigma(k)*sigma(n-k)/(k*(n-k)))/2} \\ Seiichi Manyama, Nov 09 2020

a(n) = (n-1)! * Sum_{k=1..n-1} sigma(k)*sigma(n-k)/k = (n!/2) * Sum_{k=1..n-1} sigma(k)*sigma(n-k)/(k*(n-k)). - Seiichi Manyama, Nov 09 2020.

E.g.f.: (1/2) * log( Product_{k>=1} (1 - x^k) )^2. - Ilya Gutkovskiy, Apr 24 2021

More terms from Vladeta Jovovic, Dec 28 2001

A218276 Convolution of level 2 of the divisor function.

Original entry on oeis.org

0, 0, 1, 3, 7, 16, 22, 45, 49, 100, 95, 178, 161, 304, 250, 465, 372, 676, 525, 952, 720, 1280, 946, 1702, 1217, 2156, 1570, 2764, 1925, 3376, 2360, 4185, 2912, 4944, 3404, 6121, 4047, 6960, 4858, 8344, 5530, 9600, 6391, 11246, 7513, 12496, 8372, 14926, 9486

Offset: 1

Michel Marcus, Oct 25 2012

nonn

Belongs to the family of convolution sums: Sum_{m < n*N} sigma(n)*sigma(n - N*m).
Named W2(n) by S. Alaca and K. S. Williams.
The convolution sum: Sum_{m < n} sigma(n)*sigma(n - m) = W1(n) is A000385(n+1).

G. C. Greubel, Table of n, a(n) for n = 1..1000
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, International Journal of Number Theory 3, 2 (2007), Pages 231-261.

Cf. A000385, A218277, A218278.

with(numtheory): seq((1/48)*(22*sigma[3](n) - 2*sigma[3](2*n) + 5*sigma(n) - sigma(2*n) - 24*n*sigma(n) + 6*n*sigma(2*n)),n=1..60); # Ridouane Oudra, Feb 23 2021

Table[Sum[DivisorSigma[1, k]*DivisorSigma[1, n - 2*k], {k, 1, Floor[(n - 1)/2]}], {n, 1, 50}] (* G. C. Greubel, Dec 24 2016 *)

lista(nn) = {for (i=1, nn, s = sum(m=1, floor((i-1)/2), sigma(m)*sigma(i-2*m)); print1(s , ", "););}

lista(nn) = {for (i=1, nn, v = sigma(i,3)/12 - i*sigma(i)/8 + sigma(i)/24;if (i%2 == 0, v += sigma(i/2,3)/3 - i*sigma(i/2)/4 + sigma(i/2)/24); print1(v , ", "););}

a(n) = Sum_{m < 2*n} sigma(n)*sigma(n - 2*m).
a(n) = sigma_3(n)/12 + sigma_3(n/2)/3 - n*sigma(n)/8 - n*sigma(n/2)/4 + sigma(n)/24 + sigma(n/2)/24.
a(n) = (1/48)*(22*sigma_3(n) - 2*sigma_3(2*n) + 5*sigma(n) - sigma(2*n) - 24*n*sigma(n) + 6*n*sigma(2*n)). - Ridouane Oudra, Feb 23 2021

A218277 Convolution of level 3 of the divisor function.

Original entry on oeis.org

0, 0, 0, 1, 3, 4, 10, 15, 24, 33, 45, 65, 77, 102, 143, 155, 180, 268, 255, 315, 434, 435, 462, 695, 593, 735, 960, 918, 945, 1437, 1160, 1395, 1825, 1692, 1668, 2549, 1995, 2385, 3073, 2775, 2730, 4190, 3157, 3747, 4739, 4290, 4140, 6355, 4686, 5523, 7044

Offset: 1

Michel Marcus, Oct 25 2012

nonn

Named W3(n) by S. Alaca and K. S. Williams.

Robert Israel, Table of n, a(n) for n = 1..10000
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), Pages 231-261.

Cf. A000385, A218276, A218278.

f:= n -> add(numtheory:-sigma(m)*numtheory:-sigma(n-3*m),m=1..floor((n-1)/3)):
map(f, [$1..50]); # Robert Israel, Jun 28 2018
with(numtheory): seq((1/72)*(31*sigma[3](n) - sigma[3](3*n) + 7*sigma(n) - sigma(3*n) - 30*n*sigma(n) + 6*n*sigma(3*n)), n=1..50); # Ridouane Oudra, Mar 21 2021

a[n_] := Sum[DivisorSigma[1, m] DivisorSigma[1, n-3m], {m, 1, (n-1)/3}];
Array[a, 50] (* Jean-François Alcover, Sep 19 2018 *)

lista(n) = {for (i=1, n, s = sum(m=1, floor((i-1)/3), sigma(m)*sigma(i-3*m)); print1(s , ", "););}

lista(n) = {for (i=1, n, v = sigma(i,3)/24 - i*sigma(i)/12 + sigma(i)/24;if (i%3 == 0, v += 3*sigma(i/3,3)/8 - i*sigma(i/3)/4 + sigma(i/3)/24); print1(v , ", "););}

a(n) = Sum_{m<3n} sigma(m)*sigma(n-3*m).
a(n) = sigma3(n)/24 - n*sigma(n)/12 + sigma(n)/24 + 3*sigma3(n/3)/8 - n*sigma(n/3)/4 + sigma(n/3)/24.
a(n) = (1/72)*(31*sigma_3(n) - sigma_3(3*n) + 7*sigma(n) - sigma(3*n) - 30*n*sigma(n) + 6*n*sigma(3*n)). - Ridouane Oudra, Mar 21 2021

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

cp's OEIS Frontend

A259694 a(n) = Sum(k^6*sigma(k)*sigma(n-k),k=1..n-1).

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A259695 a(n) = Sum_{k=1..n-1} k^7 * sigma(k) * sigma(n-k).

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A259696 a(n) = Sum_{k=1..n-1} k^8*sigma(k)*sigma(n-k).

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A065093 Convolution of A000010 with itself.

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A112964 Sum(mu(i)*sigma(j): i+j=n), with mu=A008683 and sigma=A000203.

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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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A059356 A diagonal of triangle in A008298.

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A218276 Convolution of level 2 of the divisor function.

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A259694 a(n) = Sum(k^6sigma(k)sigma(n-k),k=1..n-1).

A259696 a(n) = Sum_{k=1..n-1} k^8sigma(k)sigma(n-k).

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