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A191750 Dirichlet convolution of A000012 with A007947.

%I A191750 #65 Sep 08 2022 08:45:57
%S A191750 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,
%T A191750 10,40,30,72,32,11,48,54,48,35,38,60,56,42,42,96,44,60,42,72,48,36,15,
%U A191750 33,72,70,54,30,72,56,80,90,60,120,62,96,56,13,84,144,68,90,96,144,72
%N A191750 Dirichlet convolution of A000012 with A007947.
%C A191750 The squarefree kernel of n is sometimes called rad(n).
%C A191750 Sequence is multiplicative with a(p^e) = 1 + p*e.
%C A191750 Dirichlet convolution of A000005 with the function of absolute values of A097945. - _R. J. Mathar_, Jul 12 2011
%C A191750 Dirichlet convolution of phi(n)*mu(n)^2 with tau(n). - _Richard L. Ollerton_, May 07 2021
%H A191750 Seiichi Manyama, <a href="/A191750/b191750.txt">Table of n, a(n) for n = 1..10000</a> (terms 1..1837 from Vincenzo Librandi)
%H A191750 Wikipedia, <a href="http://en.wikipedia.org/wiki/Dirichlet_convolution">Dirichlet convolution</a>.
%H A191750 Wikipedia, <a href="http://en.wikipedia.org/wiki/Radical_of_an_integer">Radical of an integer</a>.
%F A191750 a(n) = Sum_{d|n} rad(d) = Sum_{d|n} A007947(d).
%F A191750 a(n) <= sigma_1(n) = A000203(n); equality holds if n is a squarefree number (A005117).
%F A191750 Dirichlet g.f.: zeta^2(s)*Product_{primes p} (1+p^(1-s)-p^(-s)). - _R. J. Mathar_, Jul 12 2011
%F A191750 G.f.: Sum_{k>=1} rad(k)*x^k/(1 - x^k). - _Ilya Gutkovskiy_, Nov 06 2018
%F A191750 a(n) = Sum_{d|n} mu(d)^2*phi(d)*tau(n/d). - _Ridouane Oudra_, Nov 19 2019
%F A191750 From _Vaclav Kotesovec_, Jun 19 2020: (Start)
%F A191750 Dirichlet g.f.: zeta(s)^2 * zeta(s-1) / zeta(2*s-2) * Product_{primes p} (1 - 1/(p^s + p)).
%F A191750 Dirichlet g.f.: zeta(s)^2 * zeta(s-1) * Product_{primes p} (1 + p^(1-2*s) - p^(2-2*s) - p^(-s)).
%F A191750 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)
%F A191750 From _Richard L. Ollerton_, May 07 2021: (Start)
%F A191750 a(n) = Sum_{k=1..n} mu(n/gcd(n,k))^2*tau(gcd(n,k)).
%F A191750 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)
%e A191750 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.
%p A191750 with(numtheory): A191750 := n -> add(ilcm(op(factorset(k))),k=divisors(n)):
%p A191750 seq(A191750(i), i=1..80); # _Peter Luschny_, Jun 23 2011
%t A191750 rad[n_]:=Times@@(FactorInteger[n][[All,1]]); A191750[n_]:=Plus@@rad/@Divisors[n]; Array[A191750,50]
%t A191750 a[1] = 1; a[n_] := Times @@ ((1 + First[#] * Last[#])& /@ FactorInteger[n]); Array[a, 100] (* _Amiram Eldar_, Sep 21 2020 *)
%o A191750 (PARI) rad(n)=local(p); p=factor(n)[, 1]; prod(i=1, length(p), p[i]);
%o A191750 A191750(n)=sumdiv(n, d, rad(d))
%o A191750 (PARI) for(n=1, 100, print1(direuler(p=2, n, (1 + p*X - X)/(1 - X)^2)[n], ", ")) \\ _Vaclav Kotesovec_, Jun 19 2020
%o A191750 (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
%Y A191750 Cf. A007947, A000012 (all 1's sequence), A005117, A073355.
%K A191750 nonn,mult,easy
%O A191750 1,2
%A A191750 _Enrique Pérez Herrero_, Jun 22 2011