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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 (list; graph; refs; listen; history; text; internal format)
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
From Zhao Hui Du, Mar 12 2024: (Start)
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)
LINKS
Zhao Hui Du, Table of n, a(n) for n = 1..4000 (first 1000 terms from Chai Wah Wu; 1001..1269 from M. F. Hasler)
Patrick De Geest, Index to related sequences.
FORMULA
a(n) = sqrt(A058412(n)). - Zak Seidov, Jul 01 2013
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^(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)}:
sort(convert(Res, list)); # Robert Israel, Feb 23 2016
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
(PARI) isok(n)={ n%10 && vecmax(digits(n^2)) < 3 } \\ Michel Marcus, Feb 24 2016, edited by M. F. Hasler, Nov 14 2017
(Magma) [n: n in [1..2*10^8 by 2] | Set(Intseq(n^2)) subset [0, 1, 2]]; // Vincenzo Librandi, Feb 24 2016
CROSSREFS
Cf. A058412 (the squares); A058412, ..., A058474 (other 3-digit combinations).
Cf. A063009, A066139. - Zak Seidov, Jul 01 2013
Sequence in context: A109830 A052035 A083144 * A134462 A156753 A118592
KEYWORD
nonn,base
AUTHOR
Patrick De Geest, Nov 15 2000
EXTENSIONS
b-file corrected by Zhao Hui Du, Mar 07 2024
STATUS
approved

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Last modified May 23 08:52 EDT 2024. Contains 372760 sequences. (Running on oeis4.)