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A069891 a(n) = Sum_{k=1..n} A007913(k), the squarefree part of k. 4
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,3

COMMENTS

Sum{1<=k<=n,k squarefree} (1/k) = Sum{1<=k<=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

Table of n, a(n) for n=0..56.

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) is the sum from d=1 to sqrt(n) of f(d)*C(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 and C(r, s) is the binomial coefficient r choose s.

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

MATHEMATICA

a[n_] := Sum[Times@@(1-#1[[1]]^2&)/@FactorInteger[d]*Binomial[Floor[n/d^2]+1, 2], {d, 1, Floor[Sqrt[n]]}]

CROSSREFS

Cf. A069087.

Sequence in context: A047705 A309839 A169799 * A190118 A249714 A250177

Adjacent sequences:  A069888 A069889 A069890 * A069892 A069893 A069894

KEYWORD

nonn

AUTHOR

Dean Hickerson, Apr 09 2002

STATUS

approved

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Last modified December 12 22:06 EST 2019. Contains 329963 sequences. (Running on oeis4.)