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A136859
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Numbers k such that k and k^2 use only the digits 0, 1, 4 and 6.
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3
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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
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OFFSET
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1,3
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COMMENTS
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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
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LINKS
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FORMULA
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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)
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EXAMPLE
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400000000000000^2 = 160000000000000000000000000000.
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MATHEMATICA
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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 *)
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CROSSREFS
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KEYWORD
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base,nonn
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AUTHOR
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Jonathan Wellons (wellons(AT)gmail.com), Jan 22 2008
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EXTENSIONS
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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
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STATUS
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approved
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