

A272678


Smallest number m such that A272677(m) = n.


2



0, 2, 5, 35, 296, 2600, 25317, 251416, 2504474, 25010000, 250044723, 2500100000, 25000316228, 250002000003, 2500004472137, 25000010000000, 250000044721361, 2500000141421358, 25000000316227767
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OFFSET

0,2


COMMENTS

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^(n3). Conjecture: a(n)/(25*10^(n3)) > 1 as n > oo.  Chai Wah Wu, May 21 2016
For odd n > 2, it seems that a(n) is about 25 * 10^(n3) + 10^(floor((n1)/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^(n3)+sqrt(10^(n1)) (for n = 0, 2, 3, 5, 6, 9, 11, 12, 15, 18, ... ) or 25*10^(n3)+sqrt(2*10^(n1)) (for n = 4, 7, 8, 14, 16, 17, ...). For n = 1, a(n) is approximately 25*10^(n3)+sqrt(3*10^(n1)) and for n = 13, a(n) is about equal to 25*10^(n3)+sqrt(4*10^(n1)). Conjecture: a(n) is always approximately to 25*10^(n3)+sqrt(k*10^(n1)) 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


LINKS

Table of n, a(n) for n=0..18.
Chai Wah Wu, Upper bounds for a(n), n = 0..1000


EXAMPLE

The smallest square beginning with 5 is 529, which has two more digits than 5, and corresponds to a(2) = 5.


CROSSREFS

Cf. A018851, A018796, A272677.
Sequence in context: A058882 A000659 A164919 * A063443 A133473 A193323
Adjacent sequences: A272675 A272676 A272677 * A272679 A272680 A272681


KEYWORD

nonn,more,base


AUTHOR

Nathan Fox, Brooke Logan, and N. J. A. Sloane, May 21 2016


EXTENSIONS

a(6)a(8) from Chai Wah Wu, May 21 2016
a(9)a(10), a(15)a(18) and corrected a(12) from Chai Wah Wu, May 22 2016
a(11)a(14) from David A. Corneth, May 22 2016


STATUS

approved



