A069891 a(n) = Sum_{k=1..n} A007913(k), the squarefree part of k.

0, 1, 3, 6, 7, 12, 18, 25, 27, 28, 38, 49, 52, 65, 79, 94, 95, 112, 114, 133, 138, 159, 181, 204, 210, 211, 237, 240, 247, 276, 306, 337, 339, 372, 406, 441, 442, 479, 517, 556, 566, 607, 649, 692, 703, 708, 754, 801, 804, 805, 807, 858, 871, 924, 930, 985, 999

Offset: 0

Author

Dean Hickerson, Apr 09 2002

Keywords

• nonn

Comments

Sum_{k=1..n, k squarefree} (1/k) = Sum{k=1..n} (mu(k)^2/k) = (1/zeta(2))*(log(n) + gamma - 2*zeta'(2)/zeta(2)) + O(1/sqrt(n)). (Suryanarayana)

References

• D. Suryanarayana, Asymptotic formula for sum_{n <= x} mu^{2}(n)/n, Indian J. Math. 9 (1967/1968) 543-545.

Links

• Amiram Eldar, Table of n, a(n) for n = 0..10000
• Rafael Jakimczuk, Two Topics in Number Theory: Sum of Divisors of the Primorial and Sum of Squarefree Parts, International Mathematical Forum, Vol. 12, 2017, no. 7, pp. 331-338.

Crossrefs

Cf. A007913, A046970, A069087.

Programs

Magma

[0] cat [&+[Squarefree(k):k in [1..n]]:n in [1..60]]; // Marius A. Burtea, Dec 19 2019

Mathematica

a[n_] := Sum[If[d == 1, 1, Times@@(1-#1[[1]]^2&) /@ FactorInteger[d]] * Binomial[Floor[n/d^2]+1, 2], {d, 1, Floor[Sqrt[n]]}]; Array[a, 100, 0] (* corrected by Amiram Eldar, Apr 02 2020 *)

PARI

a(n) = sum(k=1, n, core(k)); \\ Michel Marcus, Dec 19 2019

Formula

a(n) = Sum_{d=1..floor(sqrt(n))} f(d)*binomial(floor(n/d^2)+1, 2) where f(d)=A046970(d) is the product of 1-p^2 over all prime divisors p of d.

a(n) is asymptotic to r*n^2, where r = Pi^2/30 = 0.3289868...