A058411 Numbers k such that k^2 contains only digits {0,1,2}, not ending with zero.

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

Offset: 1

Author

Patrick De Geest, Nov 15 2000

Keywords

• nonn
• base

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.
• Hisanori Mishima, Sporadic tridigital solutions.
• OEIS Wiki, Index to sequences related to squares having only given digits.

Crossrefs

Cf. A058412 (the squares); A058412, ..., A058474 (other 3-digit combinations).

Cf. A063009, A066139. - Zak Seidov, Jul 01 2013

Programs

Magma

[n: n in [1..2*10^8 by 2] | Set(Intseq(n^2)) subset [0,1,2]]; // Vincenzo Librandi, Feb 24 2016

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 *)

PARI

isok(n)={ n%10 && vecmax(digits(n^2)) < 3 } \\ Michel Marcus, Feb 24 2016, edited by M. F. Hasler, Nov 14 2017

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

Formula

a(n) = sqrt(A058412(n)). - Zak Seidov, Jul 01 2013

Extensions

b-file corrected by Zhao Hui Du, Mar 07 2024