A136859 Numbers k such that k and k^2 use only the digits 0, 1, 4 and 6.

0, 1, 4, 10, 40, 100, 400, 1000, 4000, 10000, 40000, 100000, 400000, 1000000, 4000000, 10000000, 40000000, 100000000, 400000000, 1000000000, 4000000000, 10000000000, 40000000000, 100000000000, 400000000000, 1000000000000, 4000000000000, 10000000000000, 40000000000000, 100000000000000, 400000000000000

Offset: 1

Author

Jonathan Wellons (wellons(AT)gmail.com), Jan 22 2008

Keywords

• base
• nonn

Comments

Generated with DrScheme.

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

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

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

Examples

			400000000000000^2 = 160000000000000000000000000000.
		

Links

• David W. Wilson, Table of n, a(n) for n = 1..61
• Jonathan Wellons, Tables of Shared Digits [archived].

Crossrefs

Cf. A000290, A178501.

Programs

Mathematica

clearQ[n_]:=Module[{dc = DigitCount[n]}, dc[[2]] == dc[[3]] == dc[[5]] == dc[[7]] == dc[[8]] == dc[[9]] == 0]
Select[Range[0, 2*10^6], clearQ[#]&&clearQ[#^2] &] (* Vincenzo Librandi, Feb 02 2016 *)

Formula

Conjectures from Philippe Deléham, Mar 11 2014: (Start)

G.f.: x^2*(1+4*x)/(1-10*x^2);

a(1) = 0, a(2) = 1, a(3) = 4, a(n) = 10*a(n-2) for n>3. (End)

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

Extensions

Replaced formulas by conjectures, deleted b-file and computer program based on these conjectures. - N. J. A. Sloane, Jan 29 2016

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

Extended b-file with complete values up to a(61). - David W. Wilson, Feb 01 2016