I generally don't like "base" sequences, but when OEIS indulges in them (which it always, always does), we ought to include the B=2 case. So I present for your delectation, the binary analogue of a018851: 0,1,2,5,2,9,5,11 ... On Sun, May 22, 2016 at 9:34 AM, David Wilson <davidwwilson@comcast.net> wrote:
The last paragraph should read
r(n) <= 2*sqrt(10)*n is a tight upper bound. It is approached by numbers slightly larger than 25*10^j for large j. For example r(250044723) = 1581280251.
-----Original Message----- From: math-fun [mailto:math-fun-bounces@mailman.xmission.com] On Behalf Of David Wilson Sent: Sunday, May 22, 2016 9:28 AM To: 'math-fun' Subject: Re: [math-fun] squares beginning with n
Let r(n) be the smallest value for which r(n)^2 starts with n.
Let n >= 1. Let k(n) = floor(log10(n)) + 2 = (number of digits in n) + 1.
Then, if m(n) = ceil(sqrt(n * 10^k(n))), m(n)^2 starts with n. If r(n) >= 1 is the smallest positive integer with such that r(n)^2 starts with n, then r(n) = ceil(sqrt(n * 10^k)) for some k <= k(n).
r(n)^2 <= 40*n is a tight bound. This bound is approached by numbers slightly larger than 25*10^j, for example
-----Original Message----- From: math-fun [mailto:math-fun-bounces@mailman.xmission.com] On Behalf Of Neil Sloane Sent: Saturday, May 21, 2016 8:45 PM To: fun Subject: [math-fun] squares beginning with n
Given n, A018851 and A018796 give the smallest square that begins with n
Question: does such a square always exist, and if so how big can the smallest example be? _______________________________________________ math-fun mailing list math-fun@mailman.xmission.com https://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
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