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A342450
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a(n) is the numerator of the Schnirelmann density of the n-free numbers.
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4
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53, 157, 145, 3055, 6165, 234331, 584879, 2599496, 48785015, 292856489, 854612603, 12206236915, 8392400925, 183100803621, 1296977891119, 15258697717317, 2997253335821, 79472769236347, 556309528064071, 5960463317677243, 25033951904190895, 46938653648975843, 3099441423652148001
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OFFSET
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2,1
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COMMENTS
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k-free numbers are numbers whose exponents in their prime factorization are all less than k. E.g., the squarefree numbers (k=2, A005117), the cubefree numbers (k=3, A004709) and the biquadratefree numbers (k=4, A046100).
Let Q_k(m) be the number of k-free numbers not exceeding m. The Schnirelmann density for k-free numbers is d(k) = inf_{m>=1} Q_k(m)/m.
a(2) was found by Rogers (1964).
a(3)-a(6) were found by Orr (1969).
a(7)-a(75) were found by Hardy (1979).
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REFERENCES
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József Sándor, Dragoslav S. Mitrinovic and Borislav Crstici, Handbook of Number Theory I, Springer Science & Business Media, 2005, Chapter VI, p. 217.
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LINKS
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FORMULA
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Let d(n) = a(n)/A342451(n), and let D(n) = 1/zeta(n), the asymptotic density of the n-free numbers. Then:
Lim_{n->oo} d(n) = 1.
d(n) < D(n) (Stark, 1966).
d(n) < D(n) < d(n+1) < D(n+1) (Duncan, 1965; Erdős et al., 1978).
d(n) > 1 - Sum_{p prime} 1/p^n (Duncan, 1969).
(D(n+1)-d(n+1))/(D(n)-d(n)) < 1/2^n (Duncan, 1969).
d(n) > 1 - 1/2^n - 1/3^n - 1/5^n (Diananda and Subbarao, 1977).
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EXAMPLE
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The fractions begin with 53/88, 157/189, 145/157, 3055/3168, 6165/6272, 234331/236288, 584879/587264, 2599496/2604717, 48785015/48833536, 292856489/293001216, ...
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CROSSREFS
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KEYWORD
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nonn,frac
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AUTHOR
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STATUS
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approved
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