A259694 -id:A259694 - cp's OEIS frontend

A000441 a(n) = Sum_{k=1..n-1} ksigma(k)sigma(n-k).

0, 1, 9, 34, 95, 210, 406, 740, 1161, 1920, 2695, 4116, 5369, 7868, 9690, 13640, 16116, 22419, 25365, 34160, 38640, 50622, 55154, 73320, 77225, 100100, 107730, 135576, 141085, 182340, 184760, 233616, 243408, 297738, 301420, 385110, 377511, 467210, 478842

Offset: 1

N. J. A. Sloane

nonn

Apart from initial zero this is the convolution of A340793 and A143128. - Omar E. Pol, Feb 16 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.

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

Cf. A000385, A000477, A000499, A259692, A259693, A259694, A259695, A259696.
Cf. A000203 (sigma_1), A001158 (sigma_3), A064987.
Cf. A143128, A340793.

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(1); # N. J. A. Sloane, Jul 03 2015

a[n_] := Sum[k*DivisorSigma[1, k]*DivisorSigma[1, n-k], {k, 1, n-1}]; Array[a, 40] (* Jean-François Alcover, Feb 08 2016 *)

a(n) = sum(k=1, n-1, k*sigma(k)*sigma(n-k)); \\ Michel Marcus, Feb 02 2014

a(n) = my(f = factor(n)); ((n - 6*n^2) * sigma(f) + 5*n * sigma(f, 3)) / 24; \\ Amiram Eldar, Jan 04 2025

from sympy import divisor_sigma
def A000441(n): return (n*(1-6*n)*divisor_sigma(n)+5*n*divisor_sigma(n,3))//24 # Chai Wah Wu, Jul 25 2024

Convolution of A000203 with A064987. - Sean A. Irvine, Nov 14 2010
G.f.: x*f(x)*f'(x), where f(x) = Sum_{k>=1} k*x^k/(1 - x^k). - Ilya Gutkovskiy, Apr 28 2018
a(n) = (n/24 - n^2/4)*sigma_1(n) + (5*n/24)*sigma_3(n). - Ridouane Oudra, Sep 17 2020
Sum_{k=1..n} a(k) ~  Pi^4 * n^5 / 2160. - Vaclav Kotesovec, May 09 2022

More terms from Sean A. Irvine, Nov 14 2010

a(1)=0 prepended by Michel Marcus, Feb 02 2014

A000477 a(n) = Sum_{k=1..n-1} k^2sigma(k)sigma(n-k).

Original entry on oeis.org

0, 1, 15, 76, 275, 720, 1666, 3440, 6129, 11250, 17545, 28896, 41405, 65072, 85950, 128960, 162996, 238545, 286995, 404600, 482160, 662112, 756470, 1042560, 1150625, 1549730, 1732590, 2257920, 2443105, 3250800, 3421160, 4452096, 4791600, 6039522, 6296500

Offset: 1

N. J. A. Sloane

nonn

			G.f. = x^2 + 15*x^3 + 76*x^4 + 275*x^5 + 720*x^6 + 1666*x^7 + 3440*x^8 + ...

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.

John Cerkan, Table of n, a(n) for n = 1..10000
Jacques Touchard, On prime numbers and perfect numbers, Scripta Math., 129 (1953), 35-39. [Annotated scanned copy]

Cf. A000385, A000441, A000499, A259692, A259693, A259694, A259695, A259696.

Cf. A000203 (sigma_1), A001158 (sigma_3).

with(numtheory): 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(2); # N. J. A. Sloane, Jul 03 2015

a[n_] := Sum[k^2 DivisorSigma[1, k] DivisorSigma[1, n-k], {k, 1, n-1}]; Array[a, 35] (* Jean-François Alcover, Feb 08 2016 *)

a(n) = sum(k=1, n-1, k^2*sigma(k)*sigma(n-k)); \\ Michel Marcus, Feb 02 2014

a(n) = my(f = factor(n)); ((n^2 - 4*n^3) * sigma(f) + 3*n^2 * sigma(f, 3)) / 24; \\ Amiram Eldar, Jan 04 2025

a(n) = Sum_{k=1..n-1} k^2*sigma(k)*sigma(n-k). - Sean A. Irvine, Nov 14 2010
G.f.: x*f(x)*g'(x), where f(x) = Sum_{k>=1} k*x^k/(1 - x^k) and g(x) = Sum_{k>=1} k^2*x^k/(1 - x^k)^2. - Ilya Gutkovskiy, May 02 2018
a(n) = (n^2/24 - n^3/6)*sigma_1(n) + (n^2/8)*sigma_3(n). - Ridouane Oudra, Sep 15 2020
Sum_{k=1..n} a(k) ~ Pi^4 * n^6 / 4320. - Vaclav Kotesovec, May 09 2022

More terms from Sean A. Irvine, Nov 14 2010

a(1)=0 prepended by Michel Marcus, Feb 02 2014

A000499 a(n) = Sum_{k=1..n-1} k^3sigma(k)sigma(n-k).

Original entry on oeis.org

0, 1, 27, 184, 875, 2700, 7546, 17600, 35721, 72750, 126445, 223776, 353717, 595448, 843750, 1349120, 1827636, 2808837, 3600975, 5306000, 6667920, 9599172, 11509982, 16416000, 19015625, 26605670, 30902310, 41686848, 46948825, 64233000, 70306760, 94089216

Offset: 1

N. J. A. Sloane

nonn

			G.f. = x^2 + 27*x^3 + 184*x^4 + 875*x^5 + 2700*x^6 + 7546*x^7 + 17600*x^8 + ...

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.

John Cerkan, Table of n, a(n) for n = 1..10000
Jacques Touchard, On prime numbers and perfect numbers, Scripta Math., 129 (1953), 35-39. [Annotated scanned copy]

Cf. A000385, A000441, A000477, A259692, A259693, A259694, A259695, A259696.

Cf. A000203 (sigma_1), A001158 (sigma_3).

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(3);

a[n_] := Sum[k^3*DivisorSigma[1, k]*DivisorSigma[1, n - k], {k, 1, n - 1}]; Array[a, 32] (* Jean-François Alcover, Feb 09 2016 *)

a(n) = sum(k=1, n-1, k^3*sigma(k)*sigma(n-k)); \\ Michel Marcus, Feb 02 2014

a(n) = my(f = factor(n)); ((n^3 - 3*n^4) * sigma(f) + 2*n^3 * sigma(f, 3)) / 24; \\ Amiram Eldar, Jan 04 2025

a(n) = Sum_{k=1..n-1} k^3*sigma(k)*sigma(n-k). - Michel Marcus, Feb 02 2014
a(n) = (n^3/24 - n^4/8)*sigma_1(n) + (n^3/12)*sigma_3(n). - Ridouane Oudra, Sep 15 2020
Sum_{k=1..n} a(k) ~ Pi^4 * n^7 / 7560. - Vaclav Kotesovec, Aug 08 2022

More terms and 0 prepended by Michel Marcus, Feb 02 2014

A259692 a(n) = Sum_{k=1..n-1} k^4sigma(k)sigma(n-k).

Original entry on oeis.org

0, 1, 51, 472, 2963, 10764, 36538, 95936, 222561, 502638, 974245, 1850784, 3234269, 5826680, 8857926, 15093248, 21945012, 35369541, 48358119, 74448464, 98697648, 148971972, 187495262, 276509952, 336495665, 488970662, 590163894, 823791168, 966018241, 1358404776

Offset: 1

N. J. A. Sloane, Jul 03 2015

nonn

This was formerly A001477.

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, A259693, A259694, A259695, A259696.

Cf. A279889, A279964.

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(4);

a[n_]:=Sum[k^4*DivisorSigma[1,k]*DivisorSigma[1,n-k],{k,1,n-1}]; Table[a[n],{n,1,30}] (* Robert P. P. McKone, Sep 09 2023 *)

a(n) = sum(k=1, n-1, k^4*sigma(k)*sigma(n-k)) \\ Colin Barker, Jul 16 2015

From Ridouane Oudra, Sep 09 2023: (Start)
a(n) = (n^4/24 - n^5/10)*sigma_1(n) + (5*n^4/84)*sigma_3(n) - (691/635040)*sigma_5(n) - (13/127008)*sigma_11(n) + (691/2520)*A279889(n).
a(n) = (n^4/24 - n^5/10)*sigma_1(n) - (691/1512000 - 5*n^4/84)*sigma_3(n) - (691/756000)*sigma_7(n) + (13/72000)*sigma_11(n) - (691/3150)*A279964(n).
a(n) = (-691/1596672 + n^4/24 - n^5/10)*sigma_1(n) + (5*n^4/84)*sigma_3(n) - (691/145152 - 691*n/120960)*sigma_9(n) - (65/38016)*sigma_11(n) + (691/6048)*f(n), where f(n) = Sum_{k=1..n-1} sigma_1(k)*sigma_9(n-k). (End)

A259693 a(n) = Sum_{k=1..n-1} k^5sigma(k)sigma(n-k).

Original entry on oeis.org

0, 1, 99, 1264, 10475, 44820, 185626, 546560, 1454841, 3640950, 7868245, 16042176, 31040789, 59796968, 97525350, 177090560, 276689076, 467100189, 681356055, 1096023200, 1533162960, 2426544252, 3205401854, 4885539840, 6250705625, 9431254430, 11831779350

Offset: 1

N. J. A. Sloane, Jul 03 2015

nonn

This was formerly A001478.

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, A259694, A259695, A259696.

Cf. A279889, A279964.

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(5);

S[n_, e_] := Sum[k^e * DivisorSigma[1, k] * DivisorSigma[1, n - k], {k, 1, n - 1}]
f[e_] := Table[S[n, e], {n, 1, 27}];f[5] (* James C. McMahon, Dec 19 2023 *)

a(n) = sum(k=1, n-1, k^5*sigma(k)*sigma(n-k)) \\ Colin Barker, Jul 16 2015

From Ridouane Oudra, Dec 08 2023: (Start)
a(n) = (n^5/24 - n^6/12)*sigma_1(n) + (5*n^5/112)*sigma_3(n) - (691*n/254016)*sigma_5(n) - (65*n/254016)*sigma_11(n) + (691*n/1008)*A279889(n).
a(n) = (n^5/24 - n^6/12)*sigma_1(n) + (5*n^5/112 - 691*n/604800)*sigma_3(n) - (691*n/302400)*sigma_7(n) + (13*n/28800)*sigma_11(n) - (691*n/1260)*A279964(n).
a(n) = (-3455*n/3193344 + n^5/24 - n^6/12)*sigma_1(n) + (5*n^5/112)*sigma_3(n) + (-3455*n/290304 + 691*n^2/48384)*sigma_9(n) - (325*n/76032)*sigma_11(n) + (3455*n/12096)*f(n), where f(n) = Sum_{k=1..n-1} sigma_1(k)*sigma_9(n-k). (End)

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

cp's OEIS Frontend

A000441 a(n) = Sum_{k=1..n-1} k*sigma(k)*sigma(n-k).

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A000477 a(n) = Sum_{k=1..n-1} k^2*sigma(k)*sigma(n-k).

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A000499 a(n) = Sum_{k=1..n-1} k^3*sigma(k)*sigma(n-k).

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A259692 a(n) = Sum_{k=1..n-1} k^4*sigma(k)*sigma(n-k).

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A259693 a(n) = Sum_{k=1..n-1} k^5*sigma(k)*sigma(n-k).

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Maple

Mathematica

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Formula

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

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Programs

Maple

A000441 a(n) = Sum_{k=1..n-1} ksigma(k)sigma(n-k).

A000477 a(n) = Sum_{k=1..n-1} k^2sigma(k)sigma(n-k).

A000499 a(n) = Sum_{k=1..n-1} k^3sigma(k)sigma(n-k).

A259692 a(n) = Sum_{k=1..n-1} k^4sigma(k)sigma(n-k).

A259693 a(n) = Sum_{k=1..n-1} k^5sigma(k)sigma(n-k).

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