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.

A158523 Moebius transform of negated primes in factorization of n.

Original entry on oeis.org

1, -3, -4, 6, -6, 12, -8, -12, 12, 18, -12, -24, -14, 24, 24, 24, -18, -36, -20, -36, 32, 36, -24, 48, 30, 42, -36, -48, -30, -72, -32, -48, 48, 54, 48, 72, -38, 60, 56, 72, -42, -96, -44, -72, -72, 72, -48, -96, 56, -90, 72, -84, -54, 108, 72, 96, 80, 90, -60, 144, -62, 96, -96, 96, 84, -144, -68, -108, 96, -144
Offset: 1

Views

Author

Jaroslav Krizek, Mar 20 2009

Keywords

Examples

			a(72) = a(2^3*3^2) = (-1)^3*(2+1)*2^(3-1) * (-1)^2*(3+1)*3^(2-1) = (-12)*12 = -144.
		

Crossrefs

Programs

  • Mathematica
    f[p_, e_] := (-1)^e*(p + 1)*p^(e - 1); a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Jan 05 2023 *)
  • PARI
    a(n) = {my(f = factor(n)); prod(i = 1, #f~, (-1)^f[i,2]*(f[i,1]+1)*f[i,1]^(f[i,2]-1));} \\ Amiram Eldar, Jan 05 2023

Formula

Multiplicative with a(p^e) = (-1)^e*(p+1)*p^(e-1), e>0. a(1)=1.
a(n) = mu(n) * A061019(n) = A008683(n) * A061019(n) = A061020(n) * A007427(n) = A061020(n) * A007428(n) * A000012(n) = A007427(n) * A000012(n) * A061019(n) = A007428(n) * A000005(n) * A061019(n), where operation * denotes Dirichlet convolution. Dirichlet convolution of functions b(n), c(n) is function a(n) = b(n) * c(n) = Sum_{d|n} b(d)*c(n/d).
Inverse Moebius transform gives A061019.
a(n) = (-1)^A001222(n)*A001615(n).
Apparently the Dirichlet inverse of A048250. - R. J. Mathar, Jul 15 2010
Dirichlet g.f.: zeta(2*s-2)/(zeta(s-1)*zeta(s)). - Amiram Eldar, Jan 05 2023

Extensions

More terms from Antti Karttunen, Nov 26 2024

A178450 Dirichlet inverse of A034448 (unitary sigma).

Original entry on oeis.org

1, -3, -4, 4, -6, 12, -8, -6, 6, 18, -12, -16, -14, 24, 24, 8, -18, -18, -20, -24, 32, 36, -24, 24, 10, 42, -12, -32, -30, -72, -32, -12, 48, 54, 48, 24, -38, 60, 56, 36, -42, -96, -44, -48, -36, 72, -48, -32, 14, -30, 72, -56, -54, 36, 72, 48, 80, 90, -60, 96, -62, 96, -48, 16, 84, -144, -68, -72, 96, -144
Offset: 1

Views

Author

R. J. Mathar, Dec 22 2010

Keywords

Crossrefs

Programs

  • Haskell
    import Math.NumberTheory.Primes
    a n = product . map (\(p, e) -> if even e then 2*unPrime p^(e`div`2) else -(unPrime p+1)*unPrime p^(e`div`2)) $ factorise n -- Sebastian Karlsson, Dec 04 2021
  • Mathematica
    usigma[n_] := If[n==1, 1, Times @@ (1 + Power @@@ FactorInteger[n])];
    a[n_] := a[n] = If[n==1, 1, -Sum[usigma[n/d] a[d], {d, Most@Divisors[n]}]];
    Array[a, 70] (* Jean-François Alcover, Feb 16 2020 *)
    f[p_, e_] := If[OddQ[e], -(p+1)*p^((e-1)/2), 2*p^(e/2)]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Oct 24 2023 *)
  • PARI
    seq(n)={dirdiv(vector(n, n, n==1), vector(n, n, sumdivmult(n, d, if(gcd(d, n/d)==1, d))))} \\ Andrew Howroyd, Aug 05 2018
    
  • PARI
    A178450(n) = { my(f=factor(n)); prod(i=1, #f~, if(!(f[i,2]%2), 2*(f[i, 1]^(f[i, 2]/2)), -(1+f[i,1])*(f[i, 1]^((f[i, 2]-1)/2)))); }; \\ (After the multiplicative formula) - Antti Karttunen, Nov 26 2024
    

Formula

Dirichlet g.f.: zeta(2s-1)/(zeta(s)*zeta(s-1)). - R. J. Mathar, Apr 14 2011
Multiplicative with a(p^e) = 2*p^(e/2) if e is even, -(p+1)*p^((e-1)/2) if e is odd. - Sebastian Karlsson, Dec 04 2021

A378433 Dirichlet inverse of A325973, where A325973 is the arithmetic mean of {sum of unitary divisors} and {sum of squarefree divisors}.

Original entry on oeis.org

1, -3, -4, 5, -6, 12, -8, -9, 9, 18, -12, -20, -14, 24, 24, 15, -18, -27, -20, -30, 32, 36, -24, 36, 20, 42, -24, -40, -30, -72, -32, -27, 48, 54, 48, 42, -38, 60, 56, 54, -42, -96, -44, -60, -54, 72, -48, -60, 35, -60, 72, -70, -54, 72, 72, 72, 80, 90, -60, 120, -62, 96, -72, 45, 84, -144, -68, -90, 96, -144, -72, -72
Offset: 1

Views

Author

Antti Karttunen, Nov 26 2024

Keywords

Comments

Apparently differs from A378434 at positions given by A048111: 16, 32, 36, 48, 64, 72, 80, 81, 96, ...

Crossrefs

Programs

  • PARI
    A325973(n) = (1/2)*sumdiv(n, d, d*(issquarefree(d) + (1==gcd(d, n/d))));
    memoA378433 = Map();
    A378433(n) = if(1==n,1,my(v); if(mapisdefined(memoA378433,n,&v), v, v = -sumdiv(n,d,if(dA325973(n/d)*A378433(d),0)); mapput(memoA378433,n,v); (v)));

Formula

a(1) = 1, and for n > 1, a(n) = -Sum_{d|n, dA325973(n/d) * a(d).

A378435 Dirichlet inverse of the arithmetic mean between the Dirichlet inverses of {sum of unitary divisors} and {sum of squarefree divisors}.

Original entry on oeis.org

1, 3, 4, 4, 6, 12, 8, 6, 7, 18, 12, 16, 14, 24, 24, 9, 18, 21, 20, 24, 32, 36, 24, 24, 16, 42, 16, 32, 30, 72, 32, 15, 48, 54, 48, 25, 38, 60, 56, 36, 42, 96, 44, 48, 42, 72, 48, 36, 29, 48, 72, 56, 54, 48, 72, 48, 80, 90, 60, 96, 62, 96, 56, 24, 84, 144, 68, 72, 96, 144, 72, 33, 74, 114, 64, 80, 96, 168, 80, 54, 34
Offset: 1

Views

Author

Antti Karttunen, Nov 26 2024

Keywords

Comments

The first negative term is a(2592) = -48.
Apparently differs from A325973 at positions given by A048111: 16, 32, 36, 48, 64, 72, 80, 81, 96, ...

Crossrefs

Dirichlet inverse of A378434.

Programs

  • PARI
    A158523(n) = { my(f = factor(n)); prod(i = 1, #f~, (-1)^f[i, 2]*(f[i, 1]+1)*f[i, 1]^(f[i, 2]-1)); }; \\ From A158523
    A178450(n) = { my(f=factor(n)); prod(i=1, #f~, if(!(f[i,2]%2), 2*(f[i, 1]^(f[i, 2]/2)), -(1+f[i,1])*(f[i, 1]^((f[i, 2]-1)/2)))); };
    A378434(n) = ((A158523(n)+A178450(n))/2);
    memoA378435 = Map();
    A378435(n) = if(1==n,1,my(v); if(mapisdefined(memoA378435,n,&v), v, v = -sumdiv(n,d,if(dA378434(n/d)*A378435(d),0)); mapput(memoA378435,n,v); (v)));

Formula

a(1) = 1, and for n > 1, a(n) = -Sum_{d|n, dA378434(n/d) * a(d).
Showing 1-4 of 4 results.