A300719 -id:A300719 - cp's OEIS frontend

A300717 Möbius transform of A003557, n divided by its largest squarefree divisor.

1, 0, 0, 1, 0, 0, 0, 2, 2, 0, 0, 0, 0, 0, 0, 4, 0, 0, 0, 0, 0, 0, 0, 0, 4, 0, 6, 0, 0, 0, 0, 8, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 6, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 16, 0, 0, 0, 0, 0, 0, 0, 4, 0, 0, 0, 0, 0, 0, 0, 0, 18, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 4, 0, 0, 0, 0, 0, 0, 0, 6

Offset: 1

Antti Karttunen, Mar 11 2018

Multiplicative because A003557 is. - Andrew Howroyd, Jul 27 2018

Antti Karttunen, Table of n, a(n) for n = 1..65537

Cf. A000010, A008683, A003557, A300718, A300719, A003958.

with(numtheory): A003557 := n -> n/ilcm(op(numtheory[factorset](n))):
seq(add(mobius(d)*A003557(n/d), d in divisors(n)), n=1..100); # Ridouane Oudra, Nov 18 2019

Table[DivisorSum[n, MoebiusMu[#] EulerPhi[#] EulerPhi[n/#] &], {n, 108}] (* Michael De Vlieger, Nov 18 2019 *)
f[p_, e_] := If[e == 1, 0, (p - 1)*p^(e - 2)]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Dec 06 2022 *)

A003557(n) = { my(f=factor(n)); for (i=1, #f~, f[i, 2] = max(0,f[i, 2]-1)); factorback(f); }; \\ From A003557
A300717(n) = sumdiv(n,d,moebius(n/d)*A003557(d));

a(n) = {my(f = factor(n)); prod(i = 1, #f~, if(f[i,2] == 1, 0, (f[i,1] - 1)*f[i,1]^(f[i,2] - 2)));} \\ Amiram Eldar, Dec 06 2022

a(n) = Sum_{d|n} A008683(n/d)*A003557(d).
a(n) = A000010(n) - A300718(n).
a(n) = A003557(n) - A300719(n).
Multiplicative with a(p) = 0 and a(p^e) = (p-1)*p^(e-2) for prime p and e>1. - Werner Schulte, Sep 27 2018
Dirichlet convolution with A003958 equals A000010. - Werner Schulte, Sep 28 2018
a(n) = Sum_{d|n} mu(d)*phi(d)*phi(n/d). - Ridouane Oudra, Nov 18 2019
Dirichlet convolution of A000010 and A097945. - R. J. Mathar, Jun 02 2020
From Richard L. Ollerton, May 07 2021: (Start)
a(n) = Sum_{k=1..n} phi(gcd(n,k))*mu(gcd(n,k)).
a(n) = Sum_{k=1..n} phi(gcd(n,k))*mu(n/gcd(n,k)). (End)

A317935 Numerators of rational valued sequence whose Dirichlet convolution with itself yields A003557, n divided by largest squarefree divisor of n.

Original entry on oeis.org

1, 1, 1, 7, 1, 1, 1, 25, 11, 1, 1, 7, 1, 1, 1, 363, 1, 11, 1, 7, 1, 1, 1, 25, 19, 1, 61, 7, 1, 1, 1, 1335, 1, 1, 1, 77, 1, 1, 1, 25, 1, 1, 1, 7, 11, 1, 1, 363, 27, 19, 1, 7, 1, 61, 1, 25, 1, 1, 1, 7, 1, 1, 11, 9923, 1, 1, 1, 7, 1, 1, 1, 275, 1, 1, 19, 7, 1, 1, 1, 363, 1363, 1, 1, 7, 1, 1, 1, 25, 1, 11, 1, 7, 1, 1, 1, 1335, 1, 27, 11, 133, 1, 1, 1, 25, 1

Offset: 1

Antti Karttunen, Aug 12 2018

Multiplicative because A003557 is.

No negative terms among the first 2^20 terms. Is the sequence nonnegative?

Antti Karttunen, Table of n, a(n) for n = 1..65537
Wikipedia, Dirichlet convolution

Cf. A003557, A046644 (denominators).

Cf. also A300717, A300719, A318317.

A003557(n) = { my(f=factor(n)); for (i=1, #f~, f[i, 2]--); factorback(f); }; \\ From A003557
A317935aux(n) = if(1==n,n,(A003557(n)-sumdiv(n,d,if((d>1)&&(dA317935aux(d)*A317935aux(n/d),0)))/2);
A317935(n) = numerator(A317935aux(n));

a(n) = numerator of f(n), where f(1) = 1, f(n) = (1/2) * (A003557(n) - Sum_{d|n, d>1, d 1.

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A300717 Möbius transform of A003557, n divided by its largest squarefree divisor.

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A317935 Numerators of rational valued sequence whose Dirichlet convolution with itself yields A003557, n divided by largest squarefree divisor of n.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

PARI

Formula