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A073355 Sum of squarefree kernels of numbers <= n. 5

%I

%S 1,3,6,8,13,19,26,28,31,41,52,58,71,85,100,102,119,125,144,154,175,

%T 197,220,226,231,257,260,274,303,333,364,366,399,433,468,474,511,549,

%U 588,598,639,681,724,746,761,807,854,860,867,877,928,954,1007,1013,1068

%N Sum of squarefree kernels of numbers <= n.

%D G. Tenenbaum, "Introduction à la théorie analytique et probabiliste des nombres", Cours spécialisé, collection SMF, p. 55, 1995.

%H Vaclav Kotesovec, <a href="/A073355/b073355.txt">Table of n, a(n) for n = 1..10000</a>

%H E. Cohen, <a href="https://doi.org/10.1090/S0002-9947-1964-0166181-5">Some asymptotic formulas in the theory of numbers</a>, Trans. Amer. Math. Soc. 112 (1964) 214-227.

%H Vaclav Kotesovec, <a href="/A073355/a073355.jpg">Graph a(n)/n^2 (1000000 terms)</a>

%F a(n) = (1/2)*C*n^2 + O(n^(3/2)) where C=prod(1-1/p/(p+1))=0.7044... (see A065463). - _Benoit Cloitre_, Jan 31 2003

%F G.f.: (1/(1 - x))*Sum_{k>=1} phi(k)*mu(k)^2*x^k/(1 - x^k). - _Ilya Gutkovskiy_, Apr 15 2017

%F a(n) = Sum_{i=1..n} phi(i)*mu(i)^2*floor(n/i). - _Ridouane Oudra_, Oct 17 2019

%p with(numtheory): A073355 := n -> add(ilcm(op(factorset(k))), k = 1 .. n): seq(A073355(i), i = 1 .. 52); # _Peter Luschny_, Jun 23 2011

%t Accumulate[Table[Last[Select[Divisors[n], SquareFreeQ]], {n, 100}]] (* _Vaclav Kotesovec_, Oct 06 2016 *)

%t Drop[CoefficientList[Series[(1/(1 - x))*Sum[EulerPhi[k] MoebiusMu[k]^2*x^k/(1 - x^k), {k, 100}], {x, 0, 100}], x], 1] (* _Indranil Ghosh_, Apr 16 2017 *)

%o (PARI) print1(s=1);for(n=2,99,t=factor(n)[,1];print1(", ",s+=prod(i=1,#t,t[i]))) \\ _Charles R Greathouse IV_, Jun 24 2011

%Y Cf. A007947, A000217, A073356.

%K nonn

%O 1,2

%A _Reinhard Zumkeller_, Jul 29 2002

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Last modified July 14 15:44 EDT 2020. Contains 335729 sequences. (Running on oeis4.)