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 1-6 of 6 results.

A118206 Euler transform of the Liouville function.

Original entry on oeis.org

1, 1, 0, -1, 0, 0, 0, -1, -1, 0, 2, 0, -2, -2, 1, 2, 2, -2, -2, 0, 2, -1, -1, -2, 2, 5, 4, -5, -5, -2, 4, 2, -2, -7, 3, 8, 5, -7, -6, 1, 14, 4, -9, -14, 2, 5, 5, -10, -7, 6, 22, 3, -12, -20, 1, 15, 15, -16, -12, 4, 25, 6, -14, -31, 13, 33, 14, -39, -32, -6, 39, 15, -20, -31, 33, 41, 14, -53, -44, 3, 66, 12, -35, -51, 22, 48, 36, -60, -43, 21
Offset: 0

Views

Author

Stuart Clary, Apr 15 2006

Keywords

Links

Crossrefs

Cf. A118205, A118207, A118208, A061020, A117209.

Programs

  • Mathematica
    nmax = 100; lambda[k_Integer?Positive] := If[ k > 1, (-1)^Total[ Part[Transpose[FactorInteger[k]], 2] ], 1 ]; CoefficientList[ Series[ Product[ (1 - x^k)^(-lambda[k]), {k, 1, nmax} ], {x, 0, nmax} ], x ]
max = 100; s = Product[(1 - x^k)^(-LiouvilleLambda[k]), {k, 1, max}] + O[x]^max; CoefficientList[s, x] (* Jean-François Alcover, Dec 23 2015 *)
  • PARI
    {a(n)=polcoeff(exp(sum(m=1,n,sumdiv(m,d,d*moebius(core(d)))*x^m/m)+x*O(x^n)),n)} /* Cf. A061020 - Paul D. Hanna, Sep 22 2011 */

Formula

G.f.: A(x) = Product_{k>=1} (1 - x^k)^(-lambda(k)) where lambda(k) is the Liouville function, A008836.
Logarithmic derivative yields A061020. - Paul D. Hanna, Sep 22 2011
G.f.: A(x) = Product_{k >= 1} C(2*k,x^k), where C(k,x) denotes the k-th cyclotomic polynomial. - Peter Bala, Mar 31 2023

A118207 Expansion of Product_{k>=1} (1 + x^k)^lambda(k) where lambda(k) is the Liouville function, A008836.

Original entry on oeis.org

1, 1, -1, -2, 1, 2, 0, -2, -2, 0, 5, 2, -7, -6, 7, 9, 0, -10, -9, 4, 17, 2, -18, -12, 14, 21, 5, -26, -25, 14, 41, 4, -38, -35, 18, 53, 23, -56, -54, 31, 86, 15, -78, -85, 34, 112, 41, -110, -102, 49, 158, 40, -138, -150, 68, 195, 68, -191, -190, 69, 279, 89, -217, -253, 102, 327, 122, -336, -335, 118, 462, 142, -361, -430, 170
Offset: 0

Views

Author

Stuart Clary, Apr 15 2006

Keywords

Links

Crossrefs

Cf. A117210, A118205, A118206, A118208, A118209.

Programs

  • Mathematica
    nmax = 80; lambda[k_Integer?Positive] := If[ k > 1, (-1)^Total[ Part[Transpose[FactorInteger[k]], 2] ], 1 ]; CoefficientList[ Series[ Product[ (1 + x^k)^lambda[k], {k, 1, nmax} ], {x, 0, nmax} ], x ]
(* From version 7 on *) nmax = 80; CoefficientList[ Series[ Product[ (1 + x^k)^LiouvilleLambda[k], {k, 1, nmax}], {x, 0, nmax}], x] (* Jean-François Alcover, Jul 30 2013 *)

Formula

From Peter Bala, Apr 05 2023: (Start)
G.f.: A(x) = Product_{k >= 1} C(k,x^(2*k)) / C(k,x^k) = Product_{k >= 1} C(2*k,x^k) / C(4*k,x^k) = -Product_{k >= 1} C(k,x^(2*k)) * C(2*k,x^k), where C(k,x) denotes the k-th cyclotomic polynomial.
Conjecture: A(x^2) = Product_{k >= 1} C(k,x^k) * C(k,(-x)^k). (End)

A118205 Euler transform of the negative of the Liouville function.

Original entry on oeis.org

1, -1, 1, 0, -1, 2, -2, 2, 0, -2, 3, -2, 1, 2, -3, 3, -2, 0, 3, -2, 3, -2, 0, 2, -2, 3, -1, 0, 1, -2, 5, 0, 0, 1, -2, 1, 1, 2, 0, 1, -2, 1, 4, -1, 4, -2, -3, 6, -2, 5, 6, -8, 6, -4, 2, 9, -8, 7, -4, -1, 11, -1, 5, 1, -8, 5, 2, 4, 7, -8, 4, 2, 1, 14, -2, 0, -1, -6, 19, 2, 5, 6, -15, 12, 1, 3, 18, -17, 1, 9, 0, 29, -4, -3, 4, -13, 14, 17, 2, 0, -4
Offset: 0

Views

Author

Stuart Clary, Apr 15 2006

Keywords

Links

Crossrefs

Cf. A061020, A117208, A118206, A118207, A118208.

Programs

  • Mathematica
    nmax = 100; lambda[k_Integer?Positive] := If[ k > 1, (-1)^Total[ Part[Transpose[FactorInteger[k]], 2] ], 1 ]; CoefficientList[ Series[ Product[ (1 - x^k)^lambda[k], {k, 1, nmax} ], {x, 0, nmax} ], x ]
(* Second program (needs Mma >= 7.0): *)
nmax = 100;
Product[(1 - x^n)^LiouvilleLambda[n], {n, 1, nmax}] + O[x]^(nmax+1) // CoefficientList[#, x]& (* Jean-François Alcover, Jan 08 2020 *)

Formula

G.f.: A(x) = Product_{k>=1} (1 - x^k)^lambda(k) where lambda(k) is the Liouville function, A008836.
G.f.: A(x) = - Product_{k >= 1} C(k,x^k), where C(k,x) denotes the k-th cyclotomic polynomial. - Peter Bala, Mar 31 2023

A118209 Expansion of Sum_{k>=1} lambda(k) * k * x^k/(1 + x^k) where lambda(k) is the Liouville function, A008836.

Original entry on oeis.org

1, -3, -2, 5, -4, 6, -6, -11, 7, 12, -10, -10, -12, 18, 8, 21, -16, -21, -18, -20, 12, 30, -22, 22, 21, 36, -20, -30, -28, -24, -30, -43, 20, 48, 24, 35, -36, 54, 24, 44, -40, -36, -42, -50, -28, 66, -46, -42, 43, -63, 32, -60, -52, 60, 40, 66, 36, 84, -58, 40, -60, 90, -42, 85, 48, -60, -66, -80, 44, -72, -70, -77, -72, 108, -42
Offset: 1

Views

Author

Stuart Clary, Apr 15 2006

Keywords

Comments

Related to the logarithmic derivative of A118207(x) and A118208(x).
Related to a signed variant of A022998 via Mobius inversion. - R. J. Mathar, Jul 03 2011

Links

Crossrefs

Cf. A008836, A022998, A028260, A061019, A061020, A062157, A117212, A118207, A118208.

Programs

  • Mathematica
    nmax = 80; lambda[k_Integer?Positive] := If[ k > 1, (-1)^Total[ Part[Transpose[FactorInteger[k]], 2] ], 1 ]; Drop[ CoefficientList[ Series[ Sum[ lambda[k] k x^k/(1 + x^k), {k, 1, nmax} ], {x, 0, nmax} ], x ], 1 ]
f[p_, e_] := (p*(-p)^e+1)/(p+1); f[2, e_] := ((-1)^e*2^(e+2) - 1)/3; a[n_] := Times @@ f @@@ FactorInteger[n]; a[1] = 1; Array[a, 100] (* Amiram Eldar, Aug 12 2023 *)
  • PARI
    a(n) = sumdiv(n, d, (-1)^(n/d - 1)*(-1)^vecsum(factor(d)[,2])*d) \\ Michel Marcus, Dec 10 2016

Formula

a(n) = Sum_{d|n} (-1)^(n/d - 1)*lambda(d)*d, Dirichlet convolution of A061019 and A062157.
G.f.: A(x) is x times the logarithmic derivative of A118207(x).
G.f.: A(x) = A061020(x) - 2 A061020(x^2).
Dirichlet g.f.: zeta(s)*zeta(2s-2)*(1-2^(1-s))/zeta(s-1). - R. J. Mathar, Jul 03 2011
a(n) > 0 for n in A028260. - Michel Marcus, Dec 10 2016
Multiplicative with a(2^e) = ((-1)^e*2^(e+2) - 1)/3, and a(p^e) = (p*(-p)^e+1)/(p+1) for an odd prime p. - Amiram Eldar, Aug 12 2023

A308398 Expansion of e.g.f. exp(Sum_{k>=1} x^(k^2)*(x^(k^2) - 1)/k^2).

Original entry on oeis.org

1, -1, 3, -7, 19, -51, 61, 167, 6777, -107929, 1650691, -17839911, 157217083, -1229269627, 6185945949, -3251776921, -1151787785999, 10138302541647, 532690324952707, -14122245788830279, 443912721023736291, -7480012715591067331, 115775303074594208893, -1392396864130912381017
Offset: 0

Views

Author

Ilya Gutkovskiy, May 24 2019

Keywords

Crossrefs

Cf. A008836, A118208, A190585, A205801, A306831, A308396, A308397.

Programs

  • Mathematica
    nmax = 23; CoefficientList[Series[Exp[Sum[x^(k^2) (x^(k^2) - 1)/k^2, {k, 1, nmax}]], {x, 0, nmax}], x] Range[0, nmax]!
nmax = 23; CoefficientList[Series[Product[1/(1 + x^k)^(LiouvilleLambda[k]/k), {k, 1, nmax}], {x, 0, nmax}], x] Range[0, nmax]!

Formula

E.g.f.: Product_{k>=1} 1/(1 + x^k)^(lambda(k)/k), where lambda() is the Liouville function (A008836).

A356907 Expansion of 1 / (1 + Sum_{k>=1} lambda(k)*x^k), where lambda() is the Liouville function (A008836).

Original entry on oeis.org

1, -1, 2, -2, 2, 0, -4, 12, -22, 34, -42, 38, -6, -68, 202, -394, 616, -782, 730, -204, -1104, 3486, -6994, 11142, -14452, 14026, -5296, -17558, 60042, -123860, 201128, -266384, 268176, -124034, -273626, 1030396, -2188864, 3624290, -4898740, 5101306, -2744408
Offset: 0

Views

Author

Ilya Gutkovskiy, Oct 02 2022

Keywords

Crossrefs

Cf. A008836, A118205, A118206, A118207, A118208, A185694, A307076, A307240.

Programs

  • Mathematica
    nmax = 40; CoefficientList[Series[1/(1 + Sum[LiouvilleLambda[k] x^k, {k, 1, nmax}]), {x, 0, nmax}], x]
a[0] = 1; a[n_] := a[n] = -Sum[LiouvilleLambda[k] a[n - k], {k, 1, n}]; Table[a[n], {n, 0, 40}]

Formula

a(0) = 1; a(n) = -Sum_{k=1..n} A008836(k) * a(n-k).
Showing 1-6 of 6 results.

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