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A272678
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Smallest number m such that A272677(m) = n.
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2
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0, 2, 5, 35, 296, 2600, 25317, 251416, 2504474, 25010000, 250044723, 2500100000, 25000316228, 250002000003, 2500004472137, 25000010000000, 250000044721361, 2500000141421358, 25000000316227767
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OFFSET
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0,2
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COMMENTS
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Given n, this is the smallest number m with the property that the smallest square beginning with m has n more digits than n.
a(n) >= 25*10^(n-3). Conjecture: a(n)/(25*10^(n-3)) -> 1 as n -> oo. - Chai Wah Wu, May 21 2016
For odd n > 2, it seems that a(n) is about 25 * 10^(n-3) + 10^(floor((n-1)/2)), although a(13) breaks that pattern. - David A. Corneth, May 22 2016
Except for n = 1 and 13, a(n) appears to be approximately equal to either 25*10^(n-3)+sqrt(10^(n-1)) (for n = 0, 2, 3, 5, 6, 9, 11, 12, 15, 18, ... ) or 25*10^(n-3)+sqrt(2*10^(n-1)) (for n = 4, 7, 8, 14, 16, 17, ...). For n = 1, a(n) is approximately 25*10^(n-3)+sqrt(3*10^(n-1)) and for n = 13, a(n) is about equal to 25*10^(n-3)+sqrt(4*10^(n-1)). Conjecture: a(n) is always approximately to 25*10^(n-3)+sqrt(k*10^(n-1)) for some small integer k > 0. - Chai Wah Wu, May 22 2016
Using the above conjecture as a guide, upper bounds for a(n) can be computed (see file in links) which coincide with a(n) for n <= 19. - Chai Wah Wu, May 23 2016
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LINKS
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EXAMPLE
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The smallest square beginning with 5 is 529, which has two more digits than 5, and corresponds to a(2) = 5.
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CROSSREFS
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KEYWORD
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nonn,more,base
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AUTHOR
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EXTENSIONS
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a(9)-a(10), a(15)-a(18) and corrected a(12) from Chai Wah Wu, May 22 2016
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STATUS
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approved
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