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A342452
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a(n) is the least number at which the Schnirelmann density of the n-free numbers is attained.
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2
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176, 378, 2512, 3168, 31360, 236288, 1174528, 7814151, 48833536, 293001216, 1709645824, 12207734784, 67143319552
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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. 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. See A342450 for more information.
The value of m(k) in which Q_k(m)/m = d(k) is not necessarily unique: while for k = 2, 3 and 4 the density is attained at a single value, i.e., 176, 378 and 2512, respectively, for k = 5 the density is attained at both 3168 and 6336. Hardy (1979) found that also for k = 38, 55 and 56 the value of m(k) is not unique, and for k = 38 the density is attained in at least 3 values.
Orr (1969) proved that 5^n <= a(n) < 6^n, for n >= 5.
Diananda and Subbarao (1977) proved that the largest value of m at which the density is attained is in the interval [6^n/2, 6^n).
Hardy (1969) calculated the least value of m in this interval, for n = 2..75, but his values are not necessarily the least nor the largest.
The terms in the data section for n=2..14 were verified to be the least values. Except for n=5, they are also unique values.
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LINKS
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EXAMPLE
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The number of squarefree numbers (A005117) up to 176 is Q_2(176) = 106. It is where the Schnirelmann density inf_{m>=1} Q_2(m)/m = 106/176 = 53/88 is attained. Therefore a(2) = 176.
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CROSSREFS
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KEYWORD
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nonn,more
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AUTHOR
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STATUS
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approved
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