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

1,2

References

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

Links

Vaclav Kotesovec, Table of n, a(n) for n = 1..10000

E. 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)

Formula

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

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

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 *)

Prog

(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

Crossrefs

Cf. A007947, A000217, A073356.

Sequence in context: A337484 A139001 A090961 * A067440 A346597 A108161

Adjacent sequences: A073352 A073353 A073354 * A073356 A073357 A073358

Keyword

nonn

Author

Reinhard Zumkeller, Jul 29 2002

Status

approved