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A058411
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Numbers k such that k^2 contains only digits {0,1,2}, not ending with zero.
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11
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1, 11, 101, 149, 1001, 1011, 1101, 10001, 10011, 11001, 14499, 100001, 100011, 100101, 101001, 110001, 316261, 1000001, 1000011, 1000101, 1010001, 1010011, 1100001, 1100101, 10000001, 10000011, 10000101, 10001001, 10001011, 10001101, 10010001, 10100001, 10100011, 10110001
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OFFSET
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1,2
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COMMENTS
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Sporadic solutions (not consisting only of digits 0 and 1): a(4) = 149, a(11) = 14499, a(17) = 316261, a(209) = 4604367505011, a(715) = 10959977245460011, a(1015) = 110000500908955011, a(1665) = 10099510939154979751, ... Three infinite subsequences are given by numbers of the form 10...01, 10...011 and 110...01, but there are many others. - M. F. Hasler, Nov 14 2017
Most terms have a special pattern in that they have only digits 0 and 1 and could be written as Sum_{h=0..t} 10^x(h), where 2x(h) and x(h1)+x(h2) are distinct and x(0)=0 for the nonzero ending constraint. The number of n-digit terms in the sequence in the special pattern is A143823(n) - 2*A143823(n-1) + A143823(n-2) for n >= 2.
Terms with only digits 0 and 1 but not in the special pattern exist as well. If we define f(x) = 1 + x^768 + x^960 + x^1008 + x^1020 + x^1028 + x^1040 + x^1088 + x^1280 + x^2048, f(x)^2 is a function with all nonzero coefficients 1,2,10 (the only coefficient of x^2048 is 10 and the coefficient of x^2049 is 0). So f(10) is in the sequence but not in the special pattern. (End)
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LINKS
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FORMULA
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MAPLE
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R[1]:= {1, 9};
for m from 2 to 10 do
R[m]:= select(t -> max(convert(t^2 mod 10^m, base, 10)) <= 2, map(s -> seq(s + i*10^(m-1), i=0..9), R[m-1]))
od:
Res:= {seq(op(select(t -> t >= 10^(m-1) and max(convert(t^2, base, 10)) <= 2, R[m])), m=1..10)}:
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MATHEMATICA
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Select[Range[10^6], And[Total@ Take[RotateRight@ DigitCount@ #, -7] == 0, Mod[#, 10] != 0] &[#^2] &] (* Michael De Vlieger, Nov 14 2017 *)
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PROG
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(Python)
A058411_list = [i for i in range(10**6) if i % 10 and max(str(i**2)) < '3'] # Chai Wah Wu, Feb 23 2016
(Magma) [n: n in [1..2*10^8 by 2] | Set(Intseq(n^2)) subset [0, 1, 2]]; // Vincenzo Librandi, Feb 24 2016
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CROSSREFS
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KEYWORD
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nonn,base
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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