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A191750 Dirichlet convolution of A000012 with A007947. 1
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 (list; graph; refs; listen; history; text; internal format)
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

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

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]

PROG

(PARI) rad(n)=local(p); p=factor(n)[, 1]; prod(i=1, length(p), p[i]);

A191750(n)=sumdiv(n, d, rad(d))

(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: A299299 A103402 A154664 * A034448 A069184 A181549

Adjacent sequences:  A191747 A191748 A191749 * A191751 A191752 A191753

KEYWORD

nonn,mult,easy

AUTHOR

Enrique Pérez Herrero, Jun 22 2011

STATUS

approved

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Last modified December 18 16:49 EST 2018. Contains 318229 sequences. (Running on oeis4.)