cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-8 of 8 results.

A340793 Sequence whose partial sums give A000203.

Original entry on oeis.org

1, 2, 1, 3, -1, 6, -4, 7, -2, 5, -6, 16, -14, 10, 0, 7, -13, 21, -19, 22, -10, 4, -12, 36, -29, 11, -2, 16, -26, 42, -40, 31, -15, 6, -6, 43, -53, 22, -4, 34, -48, 54, -52, 40, -6, -6, -24, 76, -67, 36, -21, 26, -44, 66, -48, 48, -40, 10, -30, 108, -106, 34, 8
Offset: 1

Views

Author

Omar E. Pol, Jan 21 2021

Keywords

Comments

Essentially a duplicate of A053222.
Convolved with the nonzero terms of A000217 gives A175254, the volume of the stepped pyramid described in A245092.
Convolved with the nonzero terms of A046092 gives A244050, the volume of the stepped pyramid described in A244050.
Convolved with A000027 gives A024916.
Convolved with A000041 gives A138879.
Convolved with A000070 gives the nonzero terms of A066186.
Convolved with the nonzero terms of A002088 gives A086733.
Convolved with A014153 gives A182738.
Convolved with A024916 gives A000385.
Convolved with A036469 gives the nonzero terms of A277029.
Convolved with A091360 gives A276432.
Convolved with A143128 gives the nonzero terms of A000441.
For the correspondence between divisors and partitions see A336811.

Crossrefs

1 together with A053222.
Cf. A000203 (partial sums).
Cf. A000027, A000041, A000070, A000217, A000385, A000441, A002088, A002961, A014153, A024916, A036469, A046092, A053223, A053224, A053225, A053226, A053227, A066186, A086733, A091360, A138879, A143128, A175254, A182738, A237593, A244050, A245092, A276432, A277029, A336811.

Programs

  • Maple
    a:= n-> (s-> s(n)-s(n-1))(numtheory[sigma]):
seq(a(n), n=1..77);  # Alois P. Heinz, Jan 21 2021
  • Mathematica
    Join[{1}, Differences @ Table[DivisorSigma[1, n], {n, 1, 100}]] (* Amiram Eldar, Jan 21 2021 *)
  • PARI
    a(n) = if (n==1, 1, sigma(n)-sigma(n-1)); \\ Michel Marcus, Jan 22 2021

Formula

a(n) = A053222(n-1) for n>1. - Michel Marcus, Jan 22 2021

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

Views

Author

N. J. A. Sloane

Keywords

Examples

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

References

  • 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.

Links

Crossrefs

Cf. A000385, A000441, A000499, A259692, A259693, A259694, A259695, A259696.
Cf. A000203 (sigma_1), A001158 (sigma_3).

Programs

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

Formula

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

Extensions

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^3*sigma(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

Views

Author

N. J. A. Sloane

Keywords

Examples

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

References

  • 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.

Links

Crossrefs

Cf. A000385, A000441, A000477, A259692, A259693, A259694, A259695, A259696.
Cf. A000203 (sigma_1), A001158 (sigma_3).

Programs

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

Formula

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

Extensions

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

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

Views

Author

N. J. A. Sloane, Jul 03 2015

Keywords

Comments

This was formerly A001477.

Links

Crossrefs

Cf. A000385, A000441, A000477, A000499, A259693, A259694, A259695, A259696.
Cf. A279889, A279964.

Programs

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

Formula

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^5*sigma(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

Views

Author

N. J. A. Sloane, Jul 03 2015

Keywords

Comments

This was formerly A001478.

Links

Crossrefs

Cf. A000385, A000441, A000477, A000499, A259692, A259694, A259695, A259696.
Cf. A279889, A279964.

Programs

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

Formula

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)

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

Original entry on oeis.org

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

Views

Author

N. J. A. Sloane, Jul 03 2015

Keywords

Comments

This was formerly A001479.

Links

Crossrefs

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

Programs

  • Maple
    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);
  • PARI
    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

Views

Author

N. J. A. Sloane, Jul 03 2015

Keywords

Comments

This was formerly A001480.

Links

Crossrefs

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

Programs

  • Maple
    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);
  • Mathematica
    Table[Sum[k^7 DivisorSigma[1,k]DivisorSigma[1,n-k],{k,n-1}],{n,30}] (* Harvey P. Dale, Dec 14 2015 *)
  • PARI
    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^8*sigma(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

Views

Author

N. J. A. Sloane, Jul 03 2015

Keywords

Comments

This was formerly A001481.

Links

Crossrefs

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

Programs

  • Maple
    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);
  • PARI
    a(n) = sum(k=1, n-1, k^8*sigma(k)*sigma(n-k)) \\ Colin Barker, Jul 16 2015
Showing 1-8 of 8 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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