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A141492
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a(n) is the floor of the reciprocal of the difference between the 10^n-th root of 10^n and 1.
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0
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3, 21, 144, 1085, 8685, 72381, 620420, 5428680, 48254941, 434294481, 3948131653, 36191206824, 334072678386, 3102103442165, 28952965460216, 271434051189531, 2554673422960304, 24127471216847323, 228576043106974645, 2171472409516259137, 20680689614440563220
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OFFSET
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1,1
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COMMENTS
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Conjecture: This sequence converges to the number of primes < 10^n or Pi(10^n).
Define f(x) = 1/(x^(1/x) - 1). As x increases, f(x) -> 1/y - 1/2 + y/12 - y^3/720 + y^5/30240 + ... where y = log(x)/x. So if we let x = 10^n, then we have (see Formula section) a(n) = floor(f(10^n)) and, as n increases, f(10^n) = 1/((10^n)^(1/10^n) - 1) -> 10^n/(log(10)*n) - 1/2 + (log(10)/12)*n/10^n - ...
Conjecture: a(n) = floor(10^n/(log(10)*n) - 1/2) for n >= 1. (End)
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LINKS
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FORMULA
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a(n) = floor(1/((10^n)^(1/10^n) - 1)).
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PROG
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(PARI) for(x=1, n, y=1/((10^x)^(1/10^x)-1); print1(floor(y)", "))
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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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