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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
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OFFSET
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1,2
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COMMENTS
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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
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LINKS
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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.
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FORMULA
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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)
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EXAMPLE
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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.
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MAPLE
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with(numtheory): A191750 := n -> add(ilcm(op(factorset(k))), k=divisors(n)):
seq(A191750(i), i=1..80); # Peter Luschny, Jun 23 2011
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MATHEMATICA
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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 *)
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PROG
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(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
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CROSSREFS
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Cf. A007947, A000012 (all 1's sequence), A005117, A073355.
Sequence in context: A299299 A103402 A154664 * A346613 A034448 A331107
Adjacent sequences: A191747 A191748 A191749 * A191751 A191752 A191753
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KEYWORD
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nonn,mult,easy
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AUTHOR
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Enrique Pérez Herrero, Jun 22 2011
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STATUS
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approved
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