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A111499
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a(n) = floor(10^n/PrimePi(10^n)) - 1.
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1
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1, 3, 4, 7, 9, 11, 14, 16, 18, 20, 23, 25, 27, 30, 32, 34, 37, 39, 41, 44, 46, 48, 50, 53, 55, 57, 60, 62, 64, 67, 69, 71, 73, 76, 78, 80, 83, 85, 87, 90, 92, 94, 97, 99, 101, 103, 106, 108, 110, 113, 115, 117, 120, 122, 124, 126, 129, 131, 133, 136, 138, 140, 143
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OFFSET
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1,2
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COMMENTS
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PrimePi(n) is the number of primes less than or equal to n.
10^n/PrimePi(10^n) - 1) is the ratio of the number of composite numbers less than 10^n divided by the number of prime numbers less than 10^n. Conjecture: Except for the first 2 terms, the difference between successive terms is 2 or 3.
Many terms can be obtained via the following bounds by Pierre Dusart: (1 + 1/log(x)^2 + 2/log(x)^2) * x/log(x) < primepi(x) < (1 + 1/log(x)^2 + 2/log(x)^2 + 7.59/log(x)^3) * x/log(x), for x >= 88789. - Giovanni Resta, Jan 03 2020
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LINKS
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FORMULA
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MATHEMATICA
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f[n_] := Floor[10^n/PrimePi[10^n] - 1]; Table[ f[n], {n, 14}] (* Robert G. Wilson v, Nov 18 2005 *)
piB[x_] := If[x < 10^5, PrimePi[x] {1, 1}, x/Log[x] (1 + 1/Log[x] + 2/Log[x]^2 + {0, 7.59}/Log[x]^3)]; f[n_] := Floor[10^n / piB[10^n]] - 1; Reap[ Do[ If [Length[u = Union@ f@ n] > 1, Break[], Sow@ u[[1]]], {n, 1000}]][[2, 1]] (* Giovanni Resta, Jan 03 2020 *)
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PROG
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(PARI) PiRatio(m, n) = /* Good only up to 10^9 */ { local(x, p1, p2, a, b); for(x=m, n, p1=10^x; a=floor(p1/primepi(p1)-1); print1(a, ", ") ) }
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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