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A064775
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Card{ k<=n, k such that all prime divisors of k are <= sqrt(k) }.
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7
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1, 1, 1, 2, 2, 2, 2, 3, 4, 4, 4, 5, 5, 5, 5, 6, 6, 7, 7, 7, 7, 7, 7, 8, 9, 9, 10, 10, 10, 11, 11, 12, 12, 12, 12, 13, 13, 13, 13, 14, 14, 14, 14, 14, 15, 15, 15, 16, 17, 18, 18, 18, 18, 19, 19, 20, 20, 20, 20, 21, 21, 21, 22, 23, 23, 23, 23, 23, 23, 24, 24, 25, 25, 25, 26, 26, 26, 26
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OFFSET
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1,4
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COMMENTS
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A048098(n) is the n-th number k such that all prime divisors of k are <= sqrt(k).
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REFERENCES
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D. P. Parent, Exercices de théorie des nombres, Les grands classiques, Gauthier-Villars, Edition Jacques Gabay, p. 17.
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LINKS
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FORMULA
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a(n) = n - (Sum_{p<=sqrt(n)} (p-1)) - Sum_{sqrt(n)<p<=n} floor(n/p). a(n) is the largest k such that A048098(k) <= n. Asymptotically: a(n) = (1-log(2))*n + O(n/log(n)).
a(n) = n - Sum_{i=1..floor(sqrt(n))} (pi(floor(n/i)) - pi(i)).
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EXAMPLE
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Below 28, only k=27,25,24,18,16,12,9,8,4,1 have all their prime divisors less than or equal to sqrt(k), hence a(28)=10. To obtain from A048098(n): A048098(10) = 27 <= 28 < A048098(11)=30, hence a(28)=10.
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PROG
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(PARI) a(n)=n-sum(k=1, floor(sqrt(n)+10^-20), (k-1)*isprime(k))-sum(k=ceil(sqrt(n)+10^-20), n, floor(n/k)*isprime(k))
(PARI) { for (n=1, 1000, a=n - sum(k=1, floor(sqrt(n) + 10^-20), (k-1)*isprime(k)) - sum(k=ceil(sqrt(n) + 10^-20), n, floor(n/k)*isprime(k)); write("b064775.txt", n, " ", a) ) } \\ Harry J. Smith, Sep 24 2009
(Magma) [1] cat [#[k:k in [1..n]|forall{p:p in PrimeDivisors(k)| p le Sqrt(k)}]: n in [2..80]]; // Marius A. Burtea, Nov 08 2019
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CROSSREFS
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KEYWORD
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easy,nonn
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AUTHOR
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STATUS
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approved
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