A077439 Numbers k such that k and k^2 have square decimal digits.

0, 1, 10, 100, 1000, 1049, 10000, 10490, 100000, 100499, 104900, 1000000, 1004990, 1049000, 10000000, 10004999, 10049900, 10490000, 100000000, 100049990, 100499000, 104900000, 1000000000, 1000049999, 1000499900, 1004990000, 1044049999, 1049000000, 10000000000, 10000499990, 10004999000, 10049900000, 10440499990

Offset: 1

Author

Reinhard Zumkeller, Nov 06 2002

Keywords

• base
• nonn

Comments

Numbers k such that k and the square of k use only the digits 0, 1, 4 and 9.

Notice the trick used in the first Mathematica program to convert decimal numbers to base-4 numbers and then map threes into nines and then twos into fours. This saves a lot of computing. - Robert G. Wilson v, Nov 08 2002

From Robert Israel, Dec 18 2023: (Start)

If k is a term, then so is 10 * k.

Terms that do not end in 0 include

10^(2*j+1) + 5*10^j - 1 for j >= 1, and

10^(2*j+1) + 5*10^(2*j-i) - 10^(2*j-2*i) + 5*10^j - 1 for i >= 1 and j >= 4*i + 3. (End)

Examples

			1049^2 = 1100401, therefore 1049 is a term.
A046030(7)=14 is not a term, as 14^2=196 and 6 is not a square digit.
104900499999000^2 = 11004114900040199000001000000.
		

Links

• David A. Corneth, Table of n, a(n) for n = 1..466 (first 68 terms from Jonathan Wellons)
• David A. Corneth, PARI program
• David A. Corneth, pixelart from first 466 terms in b-file
• Jonathan Wellons, Tables of Shared Digits [archived].

Crossrefs

A077440(n) = a(n)^2.

Cf. A046030, A019544, A000290.

Programs

Maple

M:= 15: # for terms of up to M digits
f:= proc(n,d) n >= 10^(d-1) and convert(convert(n^2,base,10),set) subset {0,1,4,9} end proc:
g:= proc(n,d) convert(convert(n^2 mod 10^d,base,10),set) subset {0,1,4,9} end proc:
R:= 0, 1:
C:= [0,1,9]:
for d from 2 to M do
  C:= select(g,map(t -> (t, t+10^(d-1), t+4*10^(d-1), t+9*10^(d-1)), C),d);
  V:= select(f, C,d);
  R:= R, op(V);
od:
sort([R]); # Robert Israel, Dec 18 2023

Mathematica

a = {}; Do[d = FromDigits[ ReplaceAll[ IntegerDigits[n, 4], {3 -> 9, 2 -> 4}]]; If[ Union[ Join[ IntegerDigits[d^2], {0, 1, 4, 9}]] == {0, 1, 4, 9}, a = Append[a, d]], {n, 0, 3*10^5}]; a
With[{c={0,1,4,9}},Select[FromDigits/@Tuples[c,11],SubsetQ[c, IntegerDigits[ #^2]]&]] (* Requires Mathematica version 10 or later *) (* Harvey P. Dale, Jan 15 2017 *)

PARI

See PARI link

Python

from itertools import count, islice
def A077429_gen(): # generator of terms
    for m in count(0):
        s = bin(m)[2:]
        if len(s)&1: s='0'+s
        n = int(''.join({'00':'0','01':'1','10':'4','11':'9'}[s[i:i+2]] for i in range(0,len(s),2)))
        if set(str(n**2)) <= {'0','1','4','9'}:
            yield n
A077429_list = list(islice(A077429_gen(),20)) # Chai Wah Wu, Dec 19 2023

Extensions

Edited by Robert G. Wilson v, Nov 08 2002

More terms from Jonathan Wellons (wellons(AT)gmail.com), Jan 22 2008

Edited by N. J. A. Sloane, May 15 2008 at the suggestion of R. J. Mathar