%I #76 Apr 13 2024 05:02:53
%S 0,1,4,10,40,100,400,1000,4000,10000,40000,100000,400000,1000000,
%T 4000000,10000000,40000000,100000000,400000000,1000000000,4000000000,
%U 10000000000,40000000000,100000000000,400000000000,1000000000000,4000000000000,10000000000000,40000000000000,100000000000000,400000000000000
%N Numbers k such that k and k^2 use only the digits 0, 1, 4 and 6.
%C Generated with DrScheme.
%C Are the formulas conjectured or proved? For example, the analogous sequence for {0,1,2,4} contains the sporadic solution 1010000104010000101. Clearly, if a(n) is in the sequence then 10*a(n) is also in the sequence. Is there any term that is not 0, 1, or 4 times a power of 10? - _M. F. Hasler_, Jan 26 2016
%C Answer: the formulas were merely conjectures. It appears that it is an open question as to whether there is any other type of term. - _N. J. A. Sloane_, Jan 29 2016
%C _David W. Wilson_ has observed that the real number n = 2/3 = 0.66666... with n^2 = 4/9 = 0.44444... (almost) satisfies the requirement of this sequence. - _N. J. A. Sloane_, Jan 30 2016
%H David W. Wilson, <a href="/A136859/b136859.txt">Table of n, a(n) for n = 1..61</a>
%H Jonathan Wellons, <a href="https://web.archive.org/web/20090206165028/http://jonathanwellons.com/shared-digits/">Tables of Shared Digits</a> [archived].
%F Conjectures from _Philippe Deléham_, Mar 11 2014: (Start)
%F G.f.: x^2*(1+4*x)/(1-10*x^2);
%F a(1) = 0, a(2) = 1, a(3) = 4, a(n) = 10*a(n-2) for n>3. (End)
%F This yields: a(n) = 4^(n mod 2)*A178501(floor(n/2)), where A178501(n) = floor(10^(k-1)). - _M. F. Hasler_, Nov 09 2017
%e 400000000000000^2 = 160000000000000000000000000000.
%t clearQ[n_]:=Module[{dc = DigitCount[n]}, dc[[2]] == dc[[3]] == dc[[5]] == dc[[7]] == dc[[8]] == dc[[9]] == 0]
%t Select[Range[0, 2*10^6], clearQ[#]&&clearQ[#^2] &] (* _Vincenzo Librandi_, Feb 02 2016 *)
%Y Cf. A000290, A178501.
%K base,nonn
%O 1,3
%A Jonathan Wellons (wellons(AT)gmail.com), Jan 22 2008
%E Replaced formulas by conjectures, deleted b-file and computer program based on these conjectures. - _N. J. A. Sloane_, Jan 29 2016
%E _M. F. Hasler_, Jan 29 2016, reports that he has confirmed that the terms shown are complete up to a(31) = 400000000000000. - _N. J. A. Sloane_, Jan 30 2016
%E Extended b-file with complete values up to a(61). - _David W. Wilson_, Feb 01 2016