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

0,3

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.

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...

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 *)

Prog

(Magma) [0] cat [&+[Squarefree(k):k in [1..n]]:n in [1..60]]; // Marius A. Burtea, Dec 19 2019
(PARI) a(n) = sum(k=1, n, core(k)); \\ Michel Marcus, Dec 19 2019

Crossrefs

Cf. A007913, A046970, A069087.

Sequence in context: A169799 A332904 A370900 * A190118 A249714 A250177

Adjacent sequences: A069888 A069889 A069890 * A069892 A069893 A069894

Keyword

nonn

Author

Dean Hickerson, Apr 09 2002

Status

approved