

A058411


Numbers k such that k^2 contains only digits {0,1,2}, not ending with zero.


11



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

1,2


COMMENTS

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 ndigit terms in the sequence in the special pattern is A143823(n)  2*A143823(n1) + A143823(n2) 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)


LINKS



FORMULA



MAPLE

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^(m1), i=0..9), R[m1]))
od:
Res:= {seq(op(select(t > t >= 10^(m1) and max(convert(t^2, base, 10)) <= 2, R[m])), m=1..10)}:


MATHEMATICA

Select[Range[10^6], And[Total@ Take[RotateRight@ DigitCount@ #, 7] == 0, Mod[#, 10] != 0] &[#^2] &] (* Michael De Vlieger, Nov 14 2017 *)


PROG

(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


CROSSREFS



KEYWORD

nonn,base


AUTHOR



EXTENSIONS



STATUS

approved



