A378434 -id:A378434 - cp's OEIS frontend

A158523 Moebius transform of negated primes in factorization of n.

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

Jaroslav Krizek, Mar 20 2009

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

Antti Karttunen, Table of n, a(n) for n = 1..20000

Cf. A061019, A008683, A061020, A007427, A000012, A007428, A000005, A001615, A001222, A000040, A006881, A120944, A000961, A378434.

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 *)

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

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

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

R. J. Mathar, Dec 22 2010

Antti Karttunen, Table of n, a(n) for n = 1..20000 (first 1000 terms from Andrew Howroyd)

Cf. A034448, A378434.

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

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 *)

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

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

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

Antti Karttunen, Nov 26 2024

sign

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

Antti Karttunen, Table of n, a(n) for n = 1..20000

Cf. A034448, A048111, A048250, A325973, A378434.

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)));

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

Antti Karttunen, Nov 26 2024

sign

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, ...

Antti Karttunen, Table of n, a(n) for n = 1..20000

Dirichlet inverse of A378434.

Cf. A034448, A048111, A048250, A158523, A178450, A325973, A378433.

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)));

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

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A158523 Moebius transform of negated primes in factorization of n.

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A178450 Dirichlet inverse of A034448 (unitary sigma).

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

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Author

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Programs

PARI

Formula

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

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Author

Keywords

Comments

Links

Crossrefs

Programs

PARI

Formula