A073355 Sum of squarefree kernels of numbers <= n.

1, 3, 6, 8, 13, 19, 26, 28, 31, 41, 52, 58, 71, 85, 100, 102, 119, 125, 144, 154, 175, 197, 220, 226, 231, 257, 260, 274, 303, 333, 364, 366, 399, 433, 468, 474, 511, 549, 588, 598, 639, 681, 724, 746, 761, 807, 854, 860, 867, 877, 928, 954, 1007, 1013, 1068

Offset: 1

Author

Reinhard Zumkeller, Jul 29 2002

Keywords

• nonn
• easy

References

• Gérald Tenenbaum, Introduction à la théorie analytique et probabiliste des nombres, Cours spécialisé, collection SMF, 1995, p. 55.

Links

• Vaclav Kotesovec, Table of n, a(n) for n = 1..10000
• Eckford Cohen, Some asymptotic formulas in the theory of numbers, Trans. Amer. Math. Soc. 112 (1964) 214-227.
• Vaclav Kotesovec, Graph a(n)/n^2 (1000000 terms).

Crossrefs

Partial sums of A007947.

Cf. A000217, A065463, A073356.

Programs

Maple

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

Mathematica

Accumulate[Table[Last[Select[Divisors[n], SquareFreeQ]], {n, 100}]] (* Vaclav Kotesovec, Oct 06 2016 *)
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 *)
rad[n_] := Times @@ FactorInteger[n][[;; , 1]]; Accumulate[Array[rad, 100]] (* Amiram Eldar, Mar 25 2025 *)

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

Formula

a(n) = (1/2)*C*n^2 + O(n^(3/2)) where C = Product_{p prime} (1 - 1/(p*(p+1))) = 0.7044... (see A065463). - Benoit Cloitre, Jan 31 2003

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

a(n) = Sum_{i=1..n} phi(i)*mu(i)^2*floor(n/i). - Ridouane Oudra, Oct 17 2019

a(n) = Sum_{k=1..n} rad(k). - Wesley Ivan Hurt, Jun 12 2021