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.

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A321824 a(n) = Sum_{d|n, d==1 mod 4} d^8 - Sum_{d|n, d==3 mod 4} d^8.

Original entry on oeis.org

1, 1, -6560, 1, 390626, -6560, -5764800, 1, 43040161, 390626, -214358880, -6560, 815730722, -5764800, -2562506560, 1, 6975757442, 43040161, -16983563040, 390626, 37817088000, -214358880, -78310985280, -6560, 152588281251, 815730722
Offset: 1

Views

Author

N. J. A. Sloane, Nov 24 2018

Keywords

Links

Crossrefs

Column k=8 of A322143.
Cf. A321543 - A321565, A321807 - A321836 for similar sequences.
Cf. A000265.

Programs

  • Mathematica
    s[n_, r_] := DivisorSum[n, #^8 &, Mod[#, 4] == r &]; a[n_] := s[n, 1] - s[n, 3]; Array[a, 30] (* Amiram Eldar, Nov 26 2018 *)
f[p_, e_] := If[Mod[p, 4] == 1, ((p^8)^(e+1)-1)/(p^8-1), ((-p^8)^(e+1)-1)/(-p^8-1)]; f[2, e_] := 1; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 60] (* Amiram Eldar, Sep 27 2023 *)
  • PARI
    apply( A321824(n)=sumdiv(n>>valuation(n,2),d,(2-d%4)*d^8), [1..40]) \\ M. F. Hasler, Nov 26 2018

Formula

a(n) = a(A000265(n)). - M. F. Hasler, Nov 26 2018
G.f.: Sum_{k>=1} (-1)^(k-1)*(2*k - 1)^8*x^(2*k-1)/(1 - x^(2*k-1)). - Ilya Gutkovskiy, Dec 06 2018
Multiplicative with a(2^e) = 1, and for an odd prime p, ((p^8)^(e+1)-1)/(p^8-1) if p == 1 (mod 4) and ((-p^8)^(e+1)-1)/(-p^8-1) if p == 3 (mod 4). - Amiram Eldar, Sep 27 2023
a(n) = Sum_{d|n} d^8*sin(d*Pi/2). - Ridouane Oudra, Aug 17 2024

Extensions

Keyword mult from Ilya Gutkovskiy, Dec 06 2018

A321825 a(n) = Sum_{d|n, d==1 (mod 4)} d^9 - Sum_{d|n, d==3 (mod 4)} d^9.

Original entry on oeis.org

1, 1, -19682, 1, 1953126, -19682, -40353606, 1, 387400807, 1953126, -2357947690, -19682, 10604499374, -40353606, -38441425932, 1, 118587876498, 387400807, -322687697778, 1953126, 794239673292, -2357947690, -1801152661462, -19682, 3814699218751
Offset: 1

Views

Author

N. J. A. Sloane, Nov 24 2018

Keywords

Links

Crossrefs

Column k=9 of A322143.
Cf. A321543 - A321565, A321807 - A321836 for similar sequences.
Cf. A000265.

Programs

  • Mathematica
    s[n_, r_] := DivisorSum[n, #^9 &, Mod[#, 4] == r &]; a[n_] := s[n, 1] - s[n, 3]; Array[a, 30] (* Amiram Eldar, Nov 26 2018 *)
f[p_, e_] := If[Mod[p, 4] == 1, ((p^9)^(e+1)-1)/(p^9-1), ((-p^9)^(e+1)-1)/(-p^9-1)]; f[2, e_] := 1; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 60] (* Amiram Eldar, Sep 27 2023 *)
  • PARI
    apply( A321825(n)=sumdiv(n>>valuation(n,2),d,(2-d%4)*d^9), [1..40]) \\ M. F. Hasler, Nov 26 2018

Formula

a(n) = a(A000265(n)). - M. F. Hasler, Nov 26 2018
G.f.: Sum_{k>=1} (-1)^(k-1)*(2*k - 1)^9*x^(2*k-1)/(1 - x^(2*k-1)). - Ilya Gutkovskiy, Dec 22 2018
Multiplicative with a(2^e) = 1, and for an odd prime p, ((p^9)^(e+1)-1)/(p^9-1) if p == 1 (mod 4) and ((-p^9)^(e+1)-1)/(-p^9-1) if p == 3 (mod 4). - Amiram Eldar, Sep 27 2023
a(n) = Sum_{d|n} d^9*sin(d*Pi/2). - Ridouane Oudra, Aug 18 2024

A321826 a(n) = Sum_{d|n, d==1 mod 4} d^10 - Sum_{d|n, d==3 mod 4} d^10.

Original entry on oeis.org

1, 1, -59048, 1, 9765626, -59048, -282475248, 1, 3486725353, 9765626, -25937424600, -59048, 137858491850, -282475248, -576640684048, 1, 2015993900450, 3486725353, -6131066257800, 9765626, 16679598443904, -25937424600, -41426511213648, -59048, 95367441406251, 137858491850
Offset: 1

Views

Author

N. J. A. Sloane, Nov 24 2018

Keywords

Links

Crossrefs

Column k=10 of A322143.
Cf. A321543 - A321565, A321807 - A321836 for similar sequences.
Cf. A000265.

Programs

  • Mathematica
    s[n_, r_] := DivisorSum[n, #^10 &, Mod[#, 4] == r &]; a[n_] := s[n, 1] - s[n, 3]; Array[a, 30] (* Amiram Eldar, Nov 26 2018 *)
f[p_, e_] := If[Mod[p, 4] == 1, ((p^10)^(e+1)-1)/(p^10-1), ((-p^10)^(e+1)-1)/(-p^10-1)]; f[2, e_] := 1; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 60] (* Amiram Eldar, Sep 27 2023 *)
  • PARI
    apply( A321826(n)=sumdiv(n>>valuation(n,2),d,(2-d%4)*d^10), [1..40]) \\ M. F. Hasler, Nov 26 2018

Formula

a(n) = a(A000265(n)). - M. F. Hasler, Nov 26 2018
G.f.: Sum_{k>=1} (-1)^(k-1)*(2*k - 1)^10*x^(2*k-1)/(1 - x^(2*k-1)). - Ilya Gutkovskiy, Dec 22 2018
Multiplicative with a(2^e) = 1, and for an odd prime p, ((p^10)^(e+1)-1)/(p^10-1) if p == 1 (mod 4) and ((-p^10)^(e+1)-1)/(-p^10-1) if p == 3 (mod 4). - Amiram Eldar, Sep 27 2023
a(n) = Sum_{d|n} d^10*sin(d*Pi/2). - Ridouane Oudra, Sep 04 2024

A321827 a(n) = Sum_{d|n, d==1 (mod 4)} d^11 - Sum_{d|n, d==3 (mod 4)} d^11.

Original entry on oeis.org

1, 1, -177146, 1, 48828126, -177146, -1977326742, 1, 31380882463, 48828126, -285311670610, -177146, 1792160394038, -1977326742, -8649707208396, 1, 34271896307634, 31380882463, -116490258898218, 48828126, 350275523038332, -285311670610
Offset: 1

Views

Author

N. J. A. Sloane, Nov 24 2018

Keywords

Links

Crossrefs

Column k=11 of A322143.
Cf. A321543 - A321565, A321807 - A321836 for similar sequences.
Cf. A000265.

Programs

  • Mathematica
    s[n_, r_] := DivisorSum[n, #^11 &, Mod[#, 4] == r &]; a[n_] := s[n, 1] - s[n, 3]; Array[a, 30] (* Amiram Eldar, Nov 26 2018 *)
f[p_, e_] := If[Mod[p, 4] == 1, ((p^11)^(e+1)-1)/(p^11-1), ((-p^11)^(e+1)-1)/(-p^11-1)]; f[2, e_] := 1; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 60] (* Amiram Eldar, Sep 27 2023 *)
  • PARI
    apply( A321828(n)=sumdiv(n>>valuation(n,2),d,(2-d%4)*d^11), [1..40]) \\ M. F. Hasler, Nov 26 2018

Formula

a(n) = a(A000265(n)). - M. F. Hasler, Nov 26 2018
G.f.: Sum_{k>=1} (-1)^(k-1)*(2*k - 1)^11*x^(2*k-1)/(1 - x^(2*k-1)). - Ilya Gutkovskiy, Dec 06 2018
Multiplicative with a(2^e) = 1, and for an odd prime p, ((p^11)^(e+1)-1)/(p^11-1) if p == 1 (mod 4) and ((-p^11)^(e+1)-1)/(-p^11-1) if p == 3 (mod 4). - Amiram Eldar, Sep 27 2023
a(n) = Sum_{d|n} d^11*sin(d*Pi/2). - Ridouane Oudra, Sep 08 2024

A338547 a(n) = n^2 * Sum_{d|n} (-1)^(n/d + 1) * mu(d) / d^2.

Original entry on oeis.org

1, -5, 8, -12, 24, -40, 48, -48, 72, -120, 120, -96, 168, -240, 192, -192, 288, -360, 360, -288, 384, -600, 528, -384, 600, -840, 648, -576, 840, -960, 960, -768, 960, -1440, 1152, -864, 1368, -1800, 1344, -1152, 1680, -1920, 1848, -1440, 1728, -2640, 2208, -1536, 2352, -3000
Offset: 1

Views

Author

Ilya Gutkovskiy, Nov 02 2020

Keywords

Comments

Moebius transform of A162395.

Links

Crossrefs

Cf. A007434, A008683, A162395, A321543, A325596, A338548, A338549.

Programs

  • Mathematica
    Table[n^2 Sum[(-1)^(n/d + 1) MoebiusMu[d]/d^2, {d, Divisors[n]}], {n, 1, 50}]
nmax = 50; CoefficientList[Series[Sum[MoebiusMu[k] x^k (1 - x^k)/(1 + x^k)^3, {k, 1, nmax}], {x, 0, nmax}], x] // Rest
f[p_, e_] := (p^2 - 1)*p^(2*(e - 1)); f[2, 1] = -5; f[2, e_] := -3*2^(2*(e - 1)); a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Nov 15 2022 *)
  • PARI
    a(n) = n^2 * sumdiv(n, d, (-1)^(n/d+1)*moebius(d)/d^2); \\ Michel Marcus, Nov 02 2020

Formula

G.f.: Sum_{k>=1} mu(k) * x^k * (1 - x^k) / (1 + x^k)^3.
G.f. A(x) satisfies: A(x) = x * (1 - x) / (1 + x)^3 - Sum_{k>=2} A(x^k).
Dirichlet g.f.: (1 - 2^(3 - s)) * zeta(s - 2) / zeta(s).
a(n) = J_2(n) if n odd, J_2(n) - 8 * J_2(n/2) if n even, where J_2 = A007434 (Jordan function J_2).
Multiplicative with a(2) = -5, a(2^e) = -3*2^(2*(e-1)) for e > 1, and a(p^e) = (p^2-1)*p^(2*(e-1)) for p > 2. - Amiram Eldar, Nov 15 2022

A321547 a(n) = Sum_{d|n} (-1)^(d-1)*d^8.

Original entry on oeis.org

1, -255, 6562, -65791, 390626, -1673310, 5764802, -16843007, 43053283, -99609630, 214358882, -431720542, 815730722, -1470024510, 2563287812, -4311810303, 6975757442, -10978587165, 16983563042, -25699675166, 37828630724, -54661514910, 78310985282, -110523811934, 152588281251, -208011334110
Offset: 1

Views

Author

N. J. A. Sloane, Nov 23 2018

Keywords

Links

Crossrefs

Cf. A321543 - A321565, A321807 - A321836 for similar sequences.

Programs

  • Mathematica
    f[p_, e_] := (p^(8*e + 8) - 1)/(p^8 - 1); f[2, e_] := 2 - (2^(8*e + 8) - 1)/255; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 24] (* Amiram Eldar, Nov 04 2022 *)
  • PARI
    apply( a(n)=sumdiv(n, d, (-1)^(d-1)*d^8), [1..30]) \\ M. F. Hasler, Nov 26 2018

Formula

G.f.: Sum_{k>=1} (-1)^(k-1)*k^8*x^k/(1 - x^k). - Ilya Gutkovskiy, Dec 23 2018
Multiplicative with a(2^e) = 2 - (2^(8*e + 8) - 1)/255, and a(p^e) = (p^(8*e + 8) - 1)/(p^8 - 1) for p > 2. - Amiram Eldar, Nov 04 2022

A321548 a(n) = Sum_{d|n} (-1)^(d-1)*d^9.

Original entry on oeis.org

1, -511, 19684, -262655, 1953126, -10058524, 40353608, -134480383, 387440173, -998047386, 2357947692, -5170101020, 10604499374, -20620693688, 38445332184, -68853957119, 118587876498, -197981928403, 322687697780, -512998309530, 794320419872, -1204911270612, 1801152661464, -2647111858972
Offset: 1

Views

Author

N. J. A. Sloane, Nov 23 2018

Keywords

Links

Crossrefs

Cf. A321543 - A321565, A321807 - A321836 for similar sequences.

Programs

  • Mathematica
    Table[Total[(-1)^(#-1) #^9&/@Divisors[n]],{n,30}] (* Harvey P. Dale, Sep 07 2020 *)
f[p_, e_] := (p^(9*e + 9) - 1)/(p^9 - 1); f[2, e_] := 2 - (2^(9*e + 9) - 1)/511; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 25] (* Amiram Eldar, Nov 04 2022 *)
  • PARI
    apply( a(n)=sumdiv(n, d, (-1)^(d-1)*d^9), [1..30]) \\ M. F. Hasler, Nov 26 2018

Formula

G.f.: Sum_{k>=1} (-1)^(k-1)*k^9*x^k/(1 - x^k). - Ilya Gutkovskiy, Dec 23 2018
Multiplicative with a(2^e) = 2 - (2^(9*e + 9) - 1)/511, and a(p^e) = (p^(9*e + 9) - 1)/(p^9 - 1) for p > 2. - Amiram Eldar, Nov 04 2022

A321549 a(n) = Sum_{d|n} (-1)^(d-1)*d^10.

Original entry on oeis.org

1, -1023, 59050, -1049599, 9765626, -60408150, 282475250, -1074791423, 3486843451, -9990235398, 25937424602, -61978820950, 137858491850, -288972180750, 576660215300, -1100586419199, 2015993900450, -3567040850373, 6131066257802, -10249991283974, 16680163512500, -26533985367846, 41426511213650
Offset: 1

Views

Author

N. J. A. Sloane, Nov 23 2018

Keywords

Links

Crossrefs

Cf. A321543 - A321565, A321807 - A321836 for similar sequences.

Programs

  • Mathematica
    f[p_, e_] := (p^(10*e + 10) - 1)/(p^10 - 1); f[2, e_] := 2 - (2^(10*e + 10) - 1)/1023; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 24] (* Amiram Eldar, Nov 04 2022 *)
  • PARI
    apply( a(n)=sumdiv(n, d, (-1)^(d-1)*d^10), [1..30]) \\ M. F. Hasler, Nov 26 2018

Formula

G.f.: Sum_{k>=1} (-1)^(k-1)*k^10*x^k/(1 - x^k). - Ilya Gutkovskiy, Dec 23 2018
Multiplicative with a(2^e) = 2 - (2^(10*e + 10) - 1)/1023, and a(p^e) = (p^(10*e + 10) - 1)/(p^10 - 1) for p > 2. - Amiram Eldar, Nov 04 2022

A321550 a(n) = Sum_{d|n} (-1)^(d-1)*d^11.

Original entry on oeis.org

1, -2047, 177148, -4196351, 48828126, -362621956, 1977326744, -8594130943, 31381236757, -99951173922, 285311670612, -743375186948, 1792160394038, -4047587844968, 8649804864648, -17600780175359, 34271896307634, -64237391641579, 116490258898220, -204899955368226, 350279478046112
Offset: 1

Views

Author

N. J. A. Sloane, Nov 23 2018

Keywords

Links

Crossrefs

Cf. A321543 - A321565, A321807 - A321836 for similar sequences.

Programs

  • Mathematica
    f[p_, e_] := (p^(11*e + 11) - 1)/(p^11 - 1); f[2, e_] := 2 - (2^(11*e + 11) - 1)/2047; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 24] (* Amiram Eldar, Nov 04 2022 *)
  • PARI
    apply( a(n)=sumdiv(n, d, (-1)^(d-1)*d^11), [1..30]) \\ M. F. Hasler, Nov 26 2018

Formula

G.f.: Sum_{k>=1} (-1)^(k-1)*k^11*x^k/(1 - x^k). - Ilya Gutkovskiy, Dec 24 2018
Multiplicative with a(2^e) = 2 - (2^(11*e + 11) - 1)/2047, and a(p^e) = (p^(11*e + 11) - 1)/(p^11 - 1) for p > 2. - Amiram Eldar, Nov 04 2022

A372625 Expansion of Sum_{k>=1} k^2 * x^(k^2) / (1 + x^k).

Original entry on oeis.org

1, -1, 1, 3, 1, -5, 1, 3, 10, -5, 1, -6, 1, -5, 10, 19, 1, -14, 1, -13, 10, -5, 1, 10, 26, -5, 10, -13, 1, -39, 1, 19, 10, -5, 26, 14, 1, -5, 10, -6, 1, -50, 1, -13, 35, -5, 1, 46, 50, -30, 10, -13, 1, -50, 26, -30, 10, -5, 1, -11, 1, -5, 59, 83, 26, -50, 1, -13, 10, -79
Offset: 1

Views

Author

Ilya Gutkovskiy, May 07 2024

Keywords

Crossrefs

Cf. A064027, A066839, A078306, A095118, A321543, A321558, A348608.

Programs

  • Mathematica
    nmax = 70; CoefficientList[Series[Sum[k^2 x^(k^2)/(1 + x^k), {k, 1, nmax}], {x, 0, nmax}], x] // Rest
Table[DivisorSum[n, (-1)^(# + n/#) #^2 &, # <= Sqrt[n] &], {n, 1, 70}]

Formula

a(n) = Sum_{d|n, d <= sqrt(n)} (-1)^(d + n/d) * d^2.
Previous Showing 51-60 of 65 results. Next

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