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.

A117209 G.f. A(x) satisfies 1/(1-x) = Product_{k>=1} A(x^k).

Original entry on oeis.org

1, 1, 0, -1, -1, -1, 0, 0, 0, 0, 1, 0, 0, -1, 0, 1, 2, -1, -1, -2, 0, 1, 3, -1, 0, -1, 1, -1, 1, -3, 1, -1, 1, -2, 3, 0, 6, -1, -1, -6, 2, -4, 4, -3, 2, -4, 6, -5, 6, -2, 7, -5, 4, -13, 5, -3, 11, -6, 8, -14, 10, -6, 9, -14, 11, -14, 15, -13, 9, -15, 24, -13, 19, -21, 12, -20, 27, -24, 21, -26, 22, -24, 33, -33, 32, -26
Offset: 0

Views

Author

Paul D. Hanna, Mar 03 2006

Keywords

Comments

Self-convolution inverse is A117208.

Links

Crossrefs

Cf. A023900 (l.g.f.), A117208 (inverse); variants: A117210, A117211, A117212.
Cf. A008683.

Programs

  • Mathematica
    nmax = 85; CoefficientList[ Series[ Product[ (1 - x^k)^(-MoebiusMu[k]), {k, 1, nmax} ], {x, 0, nmax} ], x ] (* Stuart Clary, Apr 15 2006 *)
  • PARI
    {a(n)=polcoeff(exp(sum(k=1,n+1,sumdiv(k,d,d*moebius(d))*x^k/k)+x*O(x^n)),n)}

Formula

G.f.: A(x) = exp( Sum_{n>=1} A023900(n)*x^n/n ), where A023900 is the Dirichlet inverse of Euler totient function.
Euler transform of the Möbius function A008683. - Stuart Clary, Franklin T. Adams-Watters and Vladeta Jovovic, Apr 15 2006
G.f.: A(x) = Product_{k>=1}(1 - x^k)^(-mu(k)) where mu(k) is the Möbius function, A008683. - Stuart Clary and Franklin T. Adams-Watters, Apr 15 2006
G.f.: A(x) = Product_{k>=1} (1 + x^(2*k-1))^mu(2*k-1), where mu() is the Moebius function. - Seiichi Manyama, Jul 06 2024

A117210 G.f. A(x) satisfies (1+x) = product_{n>=1} A(x^n).

Original entry on oeis.org

1, 1, -1, -2, 0, 1, 1, 0, -1, -1, 2, 1, -2, -3, 2, 4, 2, -5, -4, 0, 5, 2, 1, -5, -1, 2, 5, -5, -2, -2, 5, -1, 3, -6, 2, 0, 11, -6, -4, -10, 12, -1, 6, -13, 5, -8, 16, -8, 9, -13, 17, -17, 7, -21, 25, -10, 22, -29, 20, -24, 34, -24, 27, -44, 35, -32, 39, -52, 45, -39, 66, -53, 47, -76, 70, -55, 79, -98, 66, -84, 115, -89
Offset: 0

Views

Author

Paul D. Hanna, Mar 03 2006

Keywords

Comments

Self-convolution inverse is A117211.

Links

Crossrefs

Cf. A117212 (l.g.f.), A117211 (inverse); variants: A117208, A117209.

Programs

  • Mathematica
    nmax = 81; CoefficientList[ Series[ Product[ (1 + x^k)^(MoebiusMu[k]), {k, 1, nmax} ], {x, 0, nmax} ], x ] (* Stuart Clary, Apr 15 2006 *)
  • PARI
    {a(n)=if(n==0,1,if(n==1,1, -polcoeff(prod(i=1,n,sum(k=0,min(n\i,n-1),a(k)*x^(i*k))+x*O(x^n)),n,x)))}

Formula

G.f.: A(x) = exp( Sum_{n>=1} A117212(n)*x^n/n ).
G.f.: A(x) = product_{k>=1}(1 + x^k)^mu(k) where mu(k) is the Möbius function, A008683 - Stuart Clary, Apr 15 2006
Weigh transform of A008683(n). - Vladeta Jovovic, Apr 20 2006

A117208 G.f. A(x) satisfies (1-x) = product_{n>=1} A(x^n).

Original entry on oeis.org

1, -1, 1, 0, 0, 1, -1, 2, -1, 1, 0, 1, 0, 1, 0, 0, 2, -1, 2, 0, 1, 0, 1, 1, 1, 0, 1, 1, 0, 2, 1, 1, 1, 0, 2, 0, 3, 0, 0, 2, 0, 3, 0, 3, -1, 2, 0, 4, 1, 1, 3, -3, 5, 1, 3, 0, 2, -1, 2, 4, 2, 4, -5, 6, -1, 2, 7, -2, 1, -1, 4, 3, 5, 2, -2, 1, 1, 8, 2, 4, -1, -3, 4, 9, 4, -2, 4, -7, 6, 7, 10, -1, -3, -1, 1, 11, 4, 8, -15, 2, 5, 7, 13, 1, -9, -7, 9
Offset: 0

Views

Author

Paul D. Hanna, Mar 03 2006

Keywords

Comments

Self-convolution inverse is A117209.

Links

Crossrefs

Cf. A023900 (l.g.f.), A117209 (inverse); variants: A117210, A117211, A117212.

Programs

  • Mathematica
    nmax = 106; CoefficientList[ Series[ Product[ (1 - x^k)^(MoebiusMu[k]), {k, 1, nmax} ], {x, 0, nmax} ], x ] (* Stuart Clary, Apr 15 2006 *)
  • PARI
    {a(n)=polcoeff(exp(-sum(k=1,n+1,sumdiv(k,d,d*moebius(d))*x^k/k)+x*O(x^n)),n)}

Formula

G.f.: A(x) = exp( -Sum_{n>=1} A023900(n)*x^n/n ), where A023900 is the Dirichlet inverse of Euler totient function.
Euler transform of the negative of the Möbius function. - Stuart Clary, Apr 15 2006
G.f.: A(x) = product_{k>=1}(1 - x^k)^mu(k) where mu(k) is the Möbius function, A008683. - Stuart Clary, Apr 15 2006

A117211 G.f. A(x) satisfies 1/(1+x) = product_{n>=1} A(x^n).

Original entry on oeis.org

1, -1, 2, -1, 1, 1, -2, 4, -4, 4, -3, 2, 0, -1, 2, -3, 4, -5, 5, -4, 4, -3, 1, 1, -2, 3, -5, 5, -5, 3, -1, 1, 3, -4, 3, -2, 2, -1, -3, 4, -6, 4, -4, 5, 0, -4, 2, -1, 4, -2, 3, -3, 6, -9, 7, -1, 1, -4, -8, 10, -6, 10, -11, 12, -9, -4, 7, -7, 15, -25, 10, -5, 13, 1, -6, 16, -21, 14, -15, 28, -6, -12, -3, 1, 18, -18, 17, -25, 13
Offset: 0

Views

Author

Paul D. Hanna, Mar 03 2006

Keywords

Comments

Self-convolution inverse is A117210.

Links

Crossrefs

Cf. A117212 (l.g.f.), A117210 (inverse); variants: A117208, A117209.

Programs

  • Mathematica
    nmax = 88; CoefficientList[ Series[ Product[ (1 + x^k)^(-MoebiusMu[k]), {k, 1, nmax} ], {x, 0, nmax} ], x ] (* Stuart Clary, Apr 15 2006 *)
  • PARI
    {a(n)=if(n==0,1,if(n==1,-1, (-1)^n-polcoeff(prod(i=1,n,sum(k=0,min(n\i,n-1),a(k)*x^(i*k))+x*O(x^n)),n,x)))}

Formula

G.f.: A(x) = exp( -Sum_{n>=1} A117212(n)*x^n/n ).
G.f.: A(x) = product_{k>=1}(1 + x^k)^(-mu(k)) where mu(k) is the Möbius function, A008683. - Stuart Clary, Apr 15 2006

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

A162728 G.f.: x/(1-x) = Sum_{n>=1} a(n)*log(1+x^n)/n.

Original entry on oeis.org

1, 3, 2, 8, 4, 6, 6, 20, 6, 12, 10, 16, 12, 18, 8, 48, 16, 18, 18, 32, 12, 30, 22, 40, 20, 36, 18, 48, 28, 24, 30, 112, 20, 48, 24, 48, 36, 54, 24, 80, 40, 36, 42, 80, 24, 66, 46, 96, 42, 60, 32, 96, 52, 54, 40, 120, 36, 84, 58, 64, 60, 90, 36, 256, 48, 60, 66, 128, 44, 72, 70
Offset: 1

Views

Author

Paul D. Hanna, Jul 12 2009

Keywords

Comments

Dirichlet inverse of A117212. - R. J. Mathar, Jul 15 2010

Examples

			x/(1-x) = log(1+x) + 3*log(1+x^2)/2 + 2*log(1+x^3)/3 + 8*log(1+x^4)/4 + 4*log(1+x^5)/5 + 6*log(1+x^6)/6 + 6*log(1+x^7)/7 + 20*log(1+x^8)/8 +...

Links

Crossrefs

Cf. A090739, A091512, A000010 (Euler phi), A220466.
Cf. A007814, A001511, A090740, A085058, A055034, A008683.

Programs

  • Maple
    nmax:=71: with(numtheory): for p from 0 to ceil(simplify(log[2](nmax))) do for n from 1 to ceil(nmax/(p+2)) do a((2*n-1)*2^p) := (p+2)*2^(p-1)*phi(2*n-1) od: od: seq(a(n), n=1..nmax); # Johannes W. Meijer, Jan 26 2013
  • Mathematica
    f[p_, e_] := (p-1)*p^(e-1); f[2, e_] := (e+2)*2^(e-1); a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Aug 27 2023 *)
  • PARI
    /* As the inverse Mobius transform of A091512: */
{a(n)=sumdiv(n,d,moebius(n/d)*valuation((2*d)^d,2))}
  • PARI
    /* From a(2n-1)=phi(2n-1); a(2n)=phi(2n)*A090739(n), we get: */
{a(n)=if(n%2==1,eulerphi(n),eulerphi(n)*valuation(3^n-1,2))}
  • PARI
    /* From x/(1-x) = Sum_{n>=1} a(n)*log(1+x^n)/n, we get: */
{a(n)=local(A=[1]);for(k=1,n,A=concat(A,0);A[ #A]=#A*(1-polcoeff(sum(m=1,#A,A[m]/m*log(1+x^m +x*O(x^#A)) ),#A)));A[n]}

Formula

a(2n-1) = phi(2n-1); a(2n) = phi(2n)*A090739(n), where A090739(n) = exponent of 2 in 3^(2n)-1.
Inverse Mobius transform of A091512, where A091512(n) = exponent of 2 in (2n)^n.
Multiplicative: a(m*n) = a(m)*a(n) when gcd(m,n)=1, with a(p) = p-1 for odd prime p and a(2)=3.
G.f.: x/(1-x)^2 = Sum_{n>=1} a(n)*x^n/(1+x^n). - Paul D. Hanna, Jul 12 2009
Dirichlet g.f.: zeta(s-1)/( zeta(s)*(1-2^(1-s)) ). - R. J. Mathar, Apr 14 2011
a((2*n-1)*2^p) = (p+2)*2^(p-1)* phi(2*n-1), p >= 0. Observe that a(2^p) = A001792(p). - Johannes W. Meijer, Jan 26 2013
Sum_{k=1..n} a(k) ~ 6*n^2 / Pi^2. - Vaclav Kotesovec, Feb 07 2019
Multiplicative with a(2^e) = (e+2)*2^(e-1) and a(p^e) = (p-1)*p^(e-1) for an odd prime p. - Amiram Eldar, Aug 27 2023
From Ridouane Oudra, Jul 05 2025: (Start)
a(n) = Sum_{k=0..A007814(n)} 2^k*phi(n/2^k).
a(n) = Sum_{d|n} mu(n/d)*d*A001511(d).
a(n) = A000010(n)*A090740(n).
a(n) = A085058(n-1)*A055034(n), for n>1. (End)
Showing 1-6 of 6 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

Content is available under The OEIS End-User License Agreement.