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

Author

Enrique Pérez Herrero, Jun 22 2011

Keywords

• nonn
• mult
• easy

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

Examples

			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.
		

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.

Crossrefs

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

Programs

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

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 *)

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

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)