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 1-3 of 3 results.

A300894 L.g.f.: log(Product_{k>=1} (1 + mu(k)^2*x^k)) = Sum_{n>=1} a(n)*x^n/n, where mu() is the Moebius function (A008683).

Original entry on oeis.org

1, 1, 4, -3, 6, 4, 8, -3, 4, 6, 12, -12, 14, 8, 24, -3, 18, 4, 20, -18, 32, 12, 24, -12, 6, 14, 4, -24, 30, 24, 32, -3, 48, 18, 48, -12, 38, 20, 56, -18, 42, 32, 44, -36, 24, 24, 48, -12, 8, 6, 72, -42, 54, 4, 72, -24, 80, 30, 60, -72, 62, 32, 32, -3, 84, 48, 68, -54, 96, 48, 72, -12, 74, 38, 24
Offset: 1

Views

Author

Ilya Gutkovskiy, Mar 14 2018

Keywords

Examples

			L.g.f.: L(x) = x + x^2/2 + 4*x^3/3 - 3*x^4/4 + 6*x^5/5 + 4*x^6/6 + 8*x^7/7 - 3*x^8/8 + 4*x^9/9 + 6*x^10/10 + ...
exp(L(x)) = 1 + x + x^2 + 2*x^3 + x^4 + 2*x^5 + 3*x^6 + 3*x^7 + 4*x^8 + 4*x^9 + 5*x^10 + ... + A087188(n)*x^n + ...

Links

Crossrefs

Cf. A005117, A008683, A048250, A065091, A087188, A300852.

Programs

  • Mathematica
    nmax = 75; Rest[CoefficientList[Series[Log[Product[(1 + MoebiusMu[k]^2 x^k), {k, 1, nmax}]], {x, 0, nmax}], x] Range[0, nmax]]
nmax = 75; Rest[CoefficientList[Series[Sum[MoebiusMu[k]^2 k x^k/(1 + x^k), {k, 1, nmax}], {x, 0, nmax}], x]]
Table[DivisorSum[n, (-1)^(n/# + 1) # &, SquareFreeQ[#] &], {n, 75}]
f[p_, e_] := If[p == 2, If[e == 1, 1, -3], p + 1]; a[1] = 1; a[n_] := Times @@ (f @@@ FactorInteger[n]); Array[a, 100] (* Amiram Eldar, Oct 25 2020 *)
  • PARI
    seq(n)=Vec(sum(k=1, n, moebius(k)^2*k*x^k/(1 + x^k) + O(x*x^n))); \\ Andrew Howroyd, Jul 20 2018
  • PARI
    a(n)={sumdiv(n, d, if(issquarefree(d), (-1)^(n/d + 1) * d))} \\ Andrew Howroyd, Jul 20 2018

Formula

G.f.: Sum_{k>=1} mu(k)^2*k*x^k/(1 + x^k).
a(n) = n + 1 if n is an odd prime (A065091).
Multiplicative with a(2^e) = 1 if e = 1, and -3 otherwise, and a(p^e) = p+1 for an odd prime p. - Amiram Eldar, Oct 25 2020
Sum_{k=1..n} a(k) ~ n^2/4. - Amiram Eldar, Nov 20 2022

Extensions

Keyword:mult added by Andrew Howroyd, Jul 20 2018

A300893 L.g.f.: log(Product_{k>=1} (1 + x^k)/(1 + x^prime(k))) = Sum_{n>=1} a(n)*x^n/n.

Original entry on oeis.org

1, -1, 1, 3, 1, 5, 1, 3, 10, 9, 1, 9, 1, 13, 16, 3, 1, 14, 1, 13, 22, 21, 1, 9, 26, 25, 37, 17, 1, 30, 1, 3, 34, 33, 36, 18, 1, 37, 40, 13, 1, 40, 1, 25, 70, 45, 1, 9, 50, 34, 52, 29, 1, 41, 56, 17, 58, 57, 1, 34, 1, 61, 94, 3, 66, 60, 1, 37, 70, 58, 1, 18, 1, 73, 116, 41, 78, 70, 1, 13, 118, 81, 1, 44, 86
Offset: 1

Views

Author

Ilya Gutkovskiy, Mar 14 2018

Keywords

Examples

			L.g.f.: L(x) = x - x^2/2 + x^3/3 + 3*x^4/4 + x^5/5 + 5*x^6/6 + x^7/7 + 3*x^8/8 + 10*x^9/9 + 9*x^10/10 + ...
exp(L(x)) = 1 + x + x^4 + x^5 + x^6 + x^7 + x^8 + 2*x^9 + 3*x^10 + ... + A096258(n)*x^n + ...

Links

Crossrefs

Cf. A006005, A018252, A023890, A096258, A300852.

Programs

  • Mathematica
    nmax = 85; Rest[CoefficientList[Series[Log[Product[(1 + x^k)/(1 + x^Prime[k]), {k, 1, nmax}]], {x, 0, nmax}], x] Range[0, nmax]]
nmax = 85; Rest[CoefficientList[Series[Sum[Boole[!PrimeQ[k]] k x^k/(1 + x^k), {k, 1, nmax}], {x, 0, nmax}], x]]
Table[DivisorSum[n, (-1)^(n/# + 1) # &, !PrimeQ[#] &], {n, 85}]

Formula

G.f.: Sum_{k>=1} A018252(k)*x^A018252(k)/(1 + x^A018252(k)).
a(n) = 1 if n is an odd prime or 1 (A006005).

A347103 G.f.: Sum_{k>=1} k * x^prime(k) / (1 + x^prime(k)).

Original entry on oeis.org

0, 1, 2, -1, 3, -1, 4, -1, 2, -2, 5, -3, 6, -3, 5, -1, 7, -1, 8, -4, 6, -4, 9, -3, 3, -5, 2, -5, 10, -4, 11, -1, 7, -6, 7, -3, 12, -7, 8, -4, 13, -5, 14, -6, 5, -8, 15, -3, 4, -2, 9, -7, 16, -1, 8, -5, 10, -9, 17, -6, 18, -10, 6, -1, 9, -6, 19, -8, 11, -6, 20, -3
Offset: 1

Views

Author

Ilya Gutkovskiy, Aug 18 2021

Keywords

Comments

a(n) is the sum of indices of prime divisors p|n such that n/p is odd, minus the sum of indices of prime divisors p|n such that n/p is even.

Links

Crossrefs

Cf. A000720, A066328, A300852, A305614, A332422.

Programs

  • Mathematica
    nmax = 72; CoefficientList[Series[Sum[k x^Prime[k]/(1 + x^Prime[k]), {k, 1, nmax}], {x, 0, nmax}], x] // Rest
Table[-DivisorSum[n, (-1)^(n/#) PrimePi[#] &, PrimeQ[#] &], {n, 1, 72}]
  • PARI
    a(n) = my(f=factor(n)[,1]); sum(k=1, #f, if ((n/f[k]) % 2, primepi(f[k]), -primepi(f[k]))); \\ Michel Marcus, Aug 19 2021

Formula

a(n) = -Sum_{p|n, p prime} (-1)^(n/p) * pi(p), where pi = A000720.
Showing 1-3 of 3 results.

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