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 1-5 of 5 results.

A332794 a(n) = Sum_{d|n} (-1)^(d + 1) * d * phi(n/d).

Original entry on oeis.org

1, -1, 5, -4, 9, -5, 13, -12, 21, -9, 21, -20, 25, -13, 45, -32, 33, -21, 37, -36, 65, -21, 45, -60, 65, -25, 81, -52, 57, -45, 61, -80, 105, -33, 117, -84, 73, -37, 125, -108, 81, -65, 85, -84, 189, -45, 93, -160, 133, -65, 165, -100, 105, -81, 189
Offset: 1

Views

Author

Ilya Gutkovskiy, Feb 24 2020

Keywords

Links

Crossrefs

Cf. A000010, A018804, A078747, A143520, A181983, A193356, A199084, A344372, A368929.

Programs

  • Mathematica
    a[n_] := Sum[(-1)^(d + 1) d EulerPhi[n/d], {d, Divisors[n]}]; Table[a[n], {n, 1, 55}]
nmax = 55; CoefficientList[Series[Sum[EulerPhi[k] x^k/(1 + x^k)^2, {k, 1, nmax}], {x, 0, nmax}], x] // Rest
a[n_] := If[OddQ[n], Sum[GCD[n, k], {k, 1, n}], Sum[(-1)^(k + 1) GCD[n, k], {k, 1, n}]]; Table[a[n], {n, 1, 55}]
f[p_, e_] := (e*(p-1) + p)*p^(e-1); f[2, e_] := -e*2^(e-1); a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Nov 04 2022 *)
  • PARI
    a(n) = sumdiv(n, d, (-1)^(d+1)*d*eulerphi(n/d)); \\ Michel Marcus, Feb 24 2020

Formula

G.f.: Sum_{k>=1} phi(k) * x^k / (1 + x^k)^2.
Dirichlet g.f.: zeta(s-1)^2 * (1 - 2^(2 - s)) / zeta(s).
a(n) = Sum_{k=1..n} gcd(n, k) if n odd, Sum_{k=1..n} (-1)^(k + 1) * gcd(n, k) if n even.
From Amiram Eldar, Nov 04 2022: (Start)
Multiplicative with a(2^e) = -e*2^(e-1), and a(p^e) = (e*(p-1) + p)*p^(e-1) for p > 2.
Sum_{k=1..n} a(k) ~ c * n^2, where c = 3*log(2)/Pi^2 = 0.210691... . (End)
a(2*n) = - Sum_{k = 1..n} gcd(2*k, n) = - A344372(n); a(2*n+1) = A018804(2*n+1). - Peter Bala, Jan 11 2024
a(n) = Sum_{k = 1..n} (-1)^(1 + gcd(k, n)) * gcd(k, n) (follows from an identity of Cesàro. See, for example, Bordelles, Lemma 1). - Peter Bala, Jan 16 2024

A326125 Expansion of Sum_{k>=1} k^2 * x^k / (1 + x^k)^2.

Original entry on oeis.org

1, 2, 12, 4, 30, 24, 56, 8, 117, 60, 132, 48, 182, 112, 360, 16, 306, 234, 380, 120, 672, 264, 552, 96, 775, 364, 1080, 224, 870, 720, 992, 32, 1584, 612, 1680, 468, 1406, 760, 2184, 240, 1722, 1344, 1892, 528, 3510, 1104, 2256, 192, 2793, 1550, 3672, 728, 2862, 2160
Offset: 1

Views

Author

Ilya Gutkovskiy, Sep 10 2019

Keywords

Links

Crossrefs

Cf. A000593, A064987, A078306, A143520, A185152, A245579, A326238, A353908.

Programs

  • Mathematica
    nmax = 54; CoefficientList[Series[Sum[k^2 x^k/(1 + x^k)^2, {k, 1, nmax}], {x, 0, nmax}], x] // Rest
Table[n Sum[(-1)^(n/d + 1) d, {d, Divisors[n]}], {n, 1, 54}]
f[p_, e_] := p^e*(p^(e+1)-1)/(p-1); f[2, e_] := 2^e; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Dec 05 2022 *)
  • PARI
    a(n)={n*sumdiv(n, d, (-1)^(n/d+1)*d)} \\ Andrew Howroyd, Sep 10 2019

Formula

G.f.: Sum_{k>=1} (-1)^(k + 1) * k * x^k * (1 + x^k) / (1 - x^k)^3.
a(n) = n * Sum_{d|n} (-1)^(n/d + 1) * d.
a(n) = n * A000593(n).
From Amiram Eldar, Dec 05 2022: (Start)
Multiplicative with a(2^e) = 2^e, and a(p^e) = p^e*(p^(e+1)-1)/(p-1) for p > 2.
Sum_{k=1..n} a(k) ~ c * n^3, where c = Pi^2/36 = 0.2741556... (A353908). (End)
Dirichlet g.f.: zeta(s-1)*zeta(s-2)*(1-2^(2-s)). - Amiram Eldar, Jan 07 2023

A326238 Expansion of Sum_{k>=1} k * x^k * (1 - x^k) / (1 + x^k)^3.

Original entry on oeis.org

1, -2, 12, -20, 30, -24, 56, -104, 117, -60, 132, -240, 182, -112, 360, -464, 306, -234, 380, -600, 672, -264, 552, -1248, 775, -364, 1080, -1120, 870, -720, 992, -1952, 1584, -612, 1680, -2340, 1406, -760, 2184, -3120, 1722, -1344, 1892, -2640, 3510, -1104, 2256
Offset: 1

Views

Author

Ilya Gutkovskiy, Sep 10 2019

Keywords

Links

Crossrefs

Cf. A002129, A064987, A113184, A143520, A185152, A326125.

Programs

  • Mathematica
    nmax = 47; CoefficientList[Series[Sum[k x^k (1 - x^k)/(1 + x^k)^3, {k, 1, nmax}], {x, 0, nmax}], x] // Rest
Table[n Sum[(-1)^(d + 1) d, {d, Divisors[n]}], {n, 1, 47}]
f[p_, e_] := p^e*(p^(e+1)-1)/(p-1); f[2, e_] := 2^e*(3-2^(e+1)); a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Dec 05 2022 *)
  • PARI
    a(n)={n*sumdiv(n, d, (-1)^(d + 1) * d)} \\ Andrew Howroyd, Sep 10 2019

Formula

G.f.: Sum_{k>=1} (-1)^(k + 1) * k^2 * x^k / (1 - x^k)^2.
a(n) = n * Sum_{d|n} (-1)^(d + 1) * d.
a(n) = n * A002129(n).
Multiplicative with a(2^e) = 2^e*(3-2^(e+1)), and a(p^e) = p^e*(p^(e+1)-1)/(p-1) if p > 2. - Amiram Eldar, Dec 05 2022
Dirichlet g.f.: zeta(s-1)*zeta(s-2)*(1-2^(3-s)). - Amiram Eldar, Jan 07 2023

A163920 Expansion of Sum_{k>0} k*(k+1)/2 * x^k / (1 - (-x)^k)^3.

Original entry on oeis.org

1, 0, 12, 9, 30, 0, 56, 60, 126, 0, 132, 126, 182, 0, 420, 316, 306, 0, 380, 330, 798, 0, 552, 888, 875, 0, 1296, 630, 870, 0, 992, 1536, 1914, 0, 2100, 1467, 1406, 0, 2652, 2360, 1722, 0, 1892, 1518, 4860, 0, 2256, 4872, 3234, 0, 4488, 2106, 2862, 0, 5060
Offset: 1

Views

Author

Paul D. Hanna, Aug 06 2009

Keywords

Links

Crossrefs

Cf. A143520 (variant), A034715.

Programs

  • Mathematica
    CoefficientList[Series[Sum[((k(k+1))/2 x^k)/(1-(-x)^k)^3,{k,100}],{x,0,100}],x] (* Harvey P. Dale, May 08 2021 *)
  • PARI
    {a(n) = if( n<1, 0, polcoeff( sum(k=1, n, k*(k+1)/2 * x^k / (1 - (-x)^k)^3, x*O(x^n)), n))}

Formula

a(4n+2) = 0.
a(2n+1) = A034715(2n+1), where A034715 is the Dirichlet convolution of triangular numbers with themselves.
a(n) = (n/4) * Sum_{d|n} (-1)^(n+d) * (d+1) * (n/d+1). - Seiichi Manyama, Jul 17 2023

A325941 Expansion of Sum_{k>=1} k * x^(2*k) / (1 + x^k)^2.

Original entry on oeis.org

0, 1, -2, 5, -4, 4, -6, 17, -14, 6, -10, 28, -12, 8, -36, 49, -16, 13, -18, 46, -52, 12, -22, 100, -44, 14, -68, 64, -28, 24, -30, 129, -84, 18, -92, 121, -36, 20, -100, 166, -40, 32, -42, 100, -192, 24, -46, 292, -90, 31, -132, 118, -52, 40, -148, 232, -148, 30, -58, 264
Offset: 1

Views

Author

Ilya Gutkovskiy, Sep 09 2019

Keywords

Links

Crossrefs

Cf. A000593, A048272, A094471, A143520.

Programs

  • Mathematica
    nmax = 60; CoefficientList[Series[Sum[k x^(2 k)/(1 + x^k)^2, {k, 1, nmax}], {x, 0, nmax}], x] // Rest
Table[Sum[(-1)^(n/d) (n - d), {d, Divisors[n]}], {n, 1, 60}]
  • PARI
    {a(n) = sumdiv(n, d, (-1)^(n/d)*(n-d))} \\ Seiichi Manyama, Sep 14 2019

Formula

G.f.: Sum_{k>=2} (-1)^k * (k - 1) * x^k / (1 - x^k)^2.
a(n) = Sum_{d|n} (-1)^(n/d) * (n - d).
a(n) = A000593(n) - n * A048272(n).
Showing 1-5 of 5 results.

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