A191750 Dirichlet convolution of A000012 with A007947.

1, 3, 4, 5, 6, 12, 8, 7, 7, 18, 12, 20, 14, 24, 24, 9, 18, 21, 20, 30, 32, 36, 24, 28, 11, 42, 10, 40, 30, 72, 32, 11, 48, 54, 48, 35, 38, 60, 56, 42, 42, 96, 44, 60, 42, 72, 48, 36, 15, 33, 72, 70, 54, 30, 72, 56, 80, 90, 60, 120, 62, 96, 56, 13, 84, 144, 68, 90, 96, 144, 72

Offset

1,2

Comments

The squarefree kernel of n is sometimes called rad(n).

Sequence is multiplicative with a(p^e) = 1 + p*e.

Dirichlet convolution of A000005 with the function of absolute values of A097945. - R. J. Mathar, Jul 12 2011

Dirichlet convolution of phi(n)*mu(n)^2 with tau(n). - Richard L. Ollerton, May 07 2021

Links

Seiichi Manyama, Table of n, a(n) for n = 1..10000 (terms 1..1837 from Vincenzo Librandi)

Wikipedia, Dirichlet convolution.

Wikipedia, Radical of an integer.

Formula

a(n) = Sum_{d|n} rad(d) = Sum_{d|n} A007947(d).

a(n) <= sigma_1(n) = A000203(n); equality holds if n is a squarefree number (A005117).

Dirichlet g.f.: zeta^2(s)*Product_{primes p} (1+p^(1-s)-p^(-s)). - R. J. Mathar, Jul 12 2011

G.f.: Sum_{k>=1} rad(k)*x^k/(1 - x^k). - Ilya Gutkovskiy, Nov 06 2018

a(n) = Sum_{d|n} mu(d)^2*phi(d)*tau(n/d). - Ridouane Oudra, Nov 19 2019

From Vaclav Kotesovec, Jun 19 2020: (Start)

Dirichlet g.f.: zeta(s)^2 * zeta(s-1) / zeta(2*s-2) * Product_{primes p} (1 - 1/(p^s + p)).

Dirichlet g.f.: zeta(s)^2 * zeta(s-1) * Product_{primes p} (1 + p^(1-2*s) - p^(2-2*s) - p^(-s)).

Sum_{k=1..n} a(k) ~ c * Pi^2 * n^2 / 12, where c = A065463 = Product_{p prime} (1 - 1/(p*(p+1))) = 0.70444220099916559... (End)

From Richard L. Ollerton, May 07 2021: (Start)

a(n) = Sum_{k=1..n} mu(n/gcd(n,k))^2*tau(gcd(n,k)).

a(n) = Sum_{k=1..n} mu(gcd(n,k))^2*tau(n/gcd(n,k))*phi(gcd(n,k))/phi(n/gcd(n,k)). (End)

Example

The divisors of 12 are 1,2,3,4,6 and 12, the squarefree kernels of these numbers are 1,2,3,2,6 and 6, so a(12) = 1+2+3+2+6+6 = 20.

Maple

with(numtheory): A191750 := n -> add(ilcm(op(factorset(k))), k=divisors(n)):
seq(A191750(i), i=1..80); # Peter Luschny, Jun 23 2011

Mathematica

rad[n_]:=Times@@(FactorInteger[n][[All, 1]]); A191750[n_]:=Plus@@rad/@Divisors[n]; Array[A191750, 50]
a[1] = 1; a[n_] := Times @@ ((1 + First[#] * Last[#])& /@ FactorInteger[n]); Array[a, 100] (* Amiram Eldar, Sep 21 2020 *)

Prog

(PARI) rad(n)=local(p); p=factor(n)[, 1]; prod(i=1, length(p), p[i]);
A191750(n)=sumdiv(n, d, rad(d))
(PARI) for(n=1, 100, print1(direuler(p=2, n, (1 + p*X - X)/(1 - X)^2)[n], ", ")) \\ Vaclav Kotesovec, Jun 19 2020
(Magma) A007947:=func< n | &*PrimeDivisors(n) >; A191750:=func< n | &+[ A007947(d): d in Divisors(n) ] >; [ A191750(n): n in [1..80] ]; // Klaus Brockhaus, Jun 27 2011

Crossrefs

Cf. A007947, A000012 (all 1's sequence), A005117, A073355.

Sequence in context: A154664 A371242 A366743 * A346613 A034448 A365211

Adjacent sequences: A191747 A191748 A191749 * A191751 A191752 A191753

Keyword

nonn,mult,easy

Author

Enrique Pérez Herrero, Jun 22 2011

Status

approved