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 1-4 of 4 results.

A344300 Expansion of Sum_{k>=1} (-1)^(k+1) * k^2 * x^(k^2) / (1 - x^(k^2)).

Original entry on oeis.org

1, 1, 1, -3, 1, 1, 1, -3, 10, 1, 1, -3, 1, 1, 1, -19, 1, 10, 1, -3, 1, 1, 1, -3, 26, 1, 10, -3, 1, 1, 1, -19, 1, 1, 1, -30, 1, 1, 1, -3, 1, 1, 1, -3, 10, 1, 1, -19, 50, 26, 1, -3, 1, 10, 1, -3, 1, 1, 1, -3, 1, 1, 10, -83, 1, 1, 1, -3, 1, 1, 1, -30, 1, 1, 26, -3, 1, 1, 1, -19
Offset: 1

Views

Author

Ilya Gutkovskiy, May 14 2021

Keywords

Comments

Excess of sum of odd squares dividing n over sum of even squares dividing n.

Links

Crossrefs

Cf. A002129, A035316, A300853, A321543, A344299.

Programs

  • Mathematica
    nmax = 80; CoefficientList[Series[Sum[(-1)^(k + 1) k^2 x^(k^2)/(1 - x^(k^2)), {k, 1, nmax}], {x, 0, nmax}], x] // Rest
Table[DivisorSum[n, (-1)^(# + 1) # &, IntegerQ[#^(1/2)] &], {n, 1, 80}]
f[p_, e_] := (p^(2*Floor[e/2] + 2) - 1)/(p^2 - 1); f[2, e_] := 2 - (2^(2*Floor[e/2] + 2) - 1)/3; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Nov 15 2022 *)
  • PARI
    a(n) = sumdiv(n, d, if (issquare(d), (-1)^((d%2)+1)*d)); \\ Michel Marcus, Aug 22 2021
  • PARI
    a(n) = {my(f = factor(n)); prod(i = 1, #f~, if(f[i,1]==2, 2 - (2^(2*floor(f[i,2]/2) + 2) - 1)/3, (f[i,1]^(2*floor(f[i,2]/2) + 2) - 1)/(f[i,1]^2 - 1)));} \\ Amiram Eldar, Nov 15 2022

Formula

Multiplicative with a(2^e) = 2 - (2^(2*floor(e/2) + 2) - 1)/3, and a(p^e) = (p^(2*floor(e/2) + 2) - 1)/(p^2 - 1) for p > 2. - Amiram Eldar, Nov 15 2022

A317529 Expansion of Sum_{k>=1} x^(k^2)/(1 + x^(k^2)).

Original entry on oeis.org

1, -1, 1, 0, 1, -1, 1, -2, 2, -1, 1, 0, 1, -1, 1, -1, 1, -2, 1, 0, 1, -1, 1, -2, 2, -1, 2, 0, 1, -1, 1, -3, 1, -1, 1, 0, 1, -1, 1, -2, 1, -1, 1, 0, 2, -1, 1, -1, 2, -2, 1, 0, 1, -2, 1, -2, 1, -1, 1, 0, 1, -1, 2, -2, 1, -1, 1, 0, 1, -1, 1, -4, 1, -1, 2, 0, 1, -1, 1, -1, 3, -1, 1, 0, 1, -1, 1, -2, 1, -2, 1, 0, 1, -1, 1
Offset: 1

Views

Author

Ilya Gutkovskiy, Jul 30 2018

Keywords

Links

Crossrefs

Cf. A000290, A010052, A046951, A048272, A113024, A300853.

Programs

  • Maple
    seq(coeff(series(add(x^(k^2)/(1+x^(k^2)),k=1..n), x,n+1),x,n),n=1..100); # Muniru A Asiru, Jul 30 2018
  • Mathematica
    nmax = 95; Rest[CoefficientList[Series[Sum[x^k^2/(1 + x^k^2), {k, 1, nmax}], {x, 0, nmax}], x]]
nmax = 95; Rest[CoefficientList[Series[Log[Product[(1 + x^k^2)^(1/k^2), {k, 1, nmax}]], {x, 0, nmax}], x] Range[0, nmax]]
Table[DivisorSum[n, (-1)^(n/# + 1) &, IntegerQ[#^(1/2)] &], {n, 95}]
f[p_, e_] := If[p == 2, If[OddQ[e], -Floor[e/2 + 1], -Floor[(e - 1)/2]], Floor[e/2 + 1]]; a[1] = 1; a[n_] := Times @@ (f @@@ FactorInteger[n]); Array[a, 100] (* Amiram Eldar, Oct 25 2020 *)
  • PARI
    A317529(n) = sumdiv(n,d,((-1)^(1+(n/d)))*issquare(d)); \\ Antti Karttunen, Nov 07 2018

Formula

G.f.: Sum_{k>=1} x^A000290(k)/(1 + x^A000290(k)).
L.g.f.: log(Product_{k>=1} (1 + x^(k^2))^(1/k^2)) = Sum_{n>=1} a(n)*x^n/n.
a(n) = Sum_{d|n} (-1)^(n/d+1)*A010052(d).
If n is odd, a(n) = A046951(n).
Multiplicative with a(2^e) = -floor(e/2+1) for odd e, -floor((e-1)/2) for even e, and a(p^e) = floor(e/2+1) for an odd prime p. - Amiram Eldar, Oct 25 2020
From Amiram Eldar, Dec 18 2023: (Start)
Dirichlet g.f.: zeta(s) * zeta(2*s) * (1 - 1/2^(s-1)).
Sum_{k=1..n} a(k) ~ c * sqrt(n), where c = -(sqrt(2)-1) * zeta(1/2) = 0.604898... (A113024). (End)

A304876 L.g.f.: log(Product_{k>=1} (1 + x^(k*(k+1)/2))) = Sum_{n>=1} a(n)*x^n/n.

Original entry on oeis.org

1, -1, 4, -1, 1, 2, 1, -1, 4, 9, 1, -10, 1, -1, 19, -1, 1, 2, 1, -11, 25, -1, 1, -10, 1, -1, 4, 27, 1, -3, 1, -1, 4, -1, 1, 26, 1, -1, 4, -11, 1, -19, 1, -1, 64, -1, 1, -10, 1, 9, 4, -1, 1, 2, 56, -29, 4, -1, 1, -35, 1, -1, 25, -1, 1, 68, 1, -1, 4, 9, 1, -46, 1, -1, 19, -1, 1
Offset: 1

Views

Author

Ilya Gutkovskiy, May 20 2018

Keywords

Examples

			L.g.f.: L(x) = x - x^2/2 + 4*x^3/3 - x^4/4 + x^5/5 + 2*x^6/6 + x^7/7 - x^8/8 + 4*x^9/9 + 9*x^10/10 + ...
exp(L(x)) = 1 + x + x^3 + x^4 + x^6 + x^7 + x^9 + 2*x^10 + x^11 + x^13 + x^14 + x^15 + 2*x^16 + ... + A024940(n)*x^n + ...

Links

Crossrefs

Cf. A000217, A010054, A024940, A185027, A300853.

Programs

  • Mathematica
    nmax = 77; Rest[CoefficientList[Series[Log[Product[1 + x^(k (k + 1)/2), {k, 1, nmax}]], {x, 0, nmax}], x] Range[0, nmax]]
nmax = 77; Rest[CoefficientList[Series[Sum[k (k + 1)/2 x^(k (k + 1)/2)/(1 + x^(k (k + 1)/2)), {k, 1, nmax}], {x, 0, nmax}], x]]
Table[DivisorSum[n, (-1)^(n/# + 1) # &, IntegerQ[(8 # + 1)^(1/2)] &], {n, 77}]
  • PARI
    A010054(n) = issquare(8*n + 1);
A304876(n) = sumdiv(n,d,(-1)^(1+(n/d)) * A010054(d)*d); \\ Antti Karttunen, Feb 20 2023

Formula

G.f.: Sum_{k>=1} (k*(k + 1)/2)*x^(k*(k+1)/2)/(1 + x^(k*(k+1)/2)).
a(n) = Sum_{d|n} (-1)^(n/d+1)*A010054(d)*d.

A304906 L.g.f.: log(Product_{k>=1} (1 + x^(k^3))) = Sum_{n>=1} a(n)*x^n/n.

Original entry on oeis.org

1, -1, 1, -1, 1, -1, 1, 7, 1, -1, 1, -1, 1, -1, 1, -9, 1, -1, 1, -1, 1, -1, 1, 7, 1, -1, 28, -1, 1, -1, 1, -9, 1, -1, 1, -1, 1, -1, 1, 7, 1, -1, 1, -1, 1, -1, 1, -9, 1, -1, 1, -1, 1, -28, 1, 7, 1, -1, 1, -1, 1, -1, 1, 55, 1, -1, 1, -1, 1, -1, 1, 7, 1, -1, 1, -1, 1, -1, 1, -9, 28, -1
Offset: 1

Views

Author

Ilya Gutkovskiy, May 20 2018

Keywords

Examples

			L.g.f.: L(x) = x - x^2/2 + x^3/3 - x^4/4 + x^5/5 - x^6/6 + x^7/7 + 7*x^8/8 + x^9/9 - x^10/10 + x^11/11 - x^12/12 + ...
exp(L(x)) = 1 + x + x^8 + x^9 + x^27 + x^28 + x^35 + x^36 + x^64 + x^65 + x^72 + x^73 + ... + A279329(n)*x^n + ...

Links

Crossrefs

Cf. A000578, A010057, A004709, A113061, A279329, A300853.

Programs

  • Mathematica
    nmax = 82; Rest[CoefficientList[Series[Log[Product[1 + x^k^3, {k, 1, nmax}]], {x, 0, nmax}], x] Range[0, nmax]]
nmax = 82; Rest[CoefficientList[Series[Sum[k^3 x^k^3/(1 + x^k^3), {k, 1, Floor[nmax^(1/3) + 1]}], {x, 0, nmax}], x]]
Table[DivisorSum[n, (-1)^(n/# + 1) # &, IntegerQ[#^(1/3)] &], {n, 82}]
f[p_, e_] := (-1)^(p+1) * (p^(3*Floor[(e+3)/3])-1)/(p^3-1) + If[p == 2 && Divisible[e, 3], 2^(e + 1), 0]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Oct 03 2023 *)
  • PARI
    seq(n)={Vec(deriv(log(prod(k=1, sqrtnint(n, 3), 1 + x^(k^3) + O(x*x^n)))))} \\ Andrew Howroyd, Jul 20 2018
  • PARI
    a(n)={sumdiv(n, d, if(n%d^3, 0, (-1)^(n/d^3 + 1) * d^3))} \\ Andrew Howroyd, Jul 20 2018

Formula

G.f.: Sum_{k>=1} k^3*x^(k^3)/(1 + x^(k^3)).
a(n) = Sum_{d|n} (-1)^(n/d+1)*A010057(d)*d.
a(n) = 1 if n is an odd cubefree.
a(n) = -1 if n is an even cubefree.
Multiplicative with a(p^e) = (-1)^(p+1) * (p^(3*floor((e+3)/3))-1)/(p^3-1) + if(p = 2 and e == 0 (mod 3), 2^(e + 1) otherwise 0). - Amiram Eldar, Oct 03 2023

Extensions

Keyword:mult added by Andrew Howroyd, Jul 20 2018
Showing 1-4 of 4 results.

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