A224955 Numbers that are not squares, but can become squares by prepending or appending one additional digit.

2, 3, 5, 6, 8, 10, 12, 14, 19, 21, 22, 24, 28, 29, 32, 40, 41, 44, 48, 52, 56, 57, 61, 62, 67, 69, 72, 76, 78, 84, 89, 90, 96, 102, 108, 115, 116, 122, 129, 136, 152, 156, 160, 168, 176, 184, 193, 202, 209, 211, 216, 220, 230, 240, 241, 249, 250, 260, 270, 280

Offset: 1

Author

Kevin L. Schwartz and Christian N. K. Anderson, Apr 21 2013

Keywords

• nonn
• base

Comments

There are potentially 15 ways for each number to become a square--by prepending a digit between 1 and 9, or appending one of {0,1,4,5,6,9}. However, only 74 of the first 10000 terms can become a square in more than one way.

Examples

			a(4)=6 because, though 6 is not a square, it can become a square by prepending a 1 to become 16. We can also obtain 36 and 64.
		

Links

• Christian N. K. Anderson, Table of n, a(n) for n = 1..10000
• Christian N. K. Anderson, List of squares that can be formed by concatenating one digit to the first 10000 terms.
• Christian N. K. Anderson, Ulam spiral of a(n), with brighter colors corresponding to the number of ways a term may become a square.

Crossrefs

Cf. A023110, A000290.

Programs

Maple

isA224955 := proc(n)
    local p,ndgs;
    if issqr(n) then
        return false;
    else
        ndgs := convert(n,base,10) ;
        for p from 1 to 9 do
            [op(ndgs),p] ;
            add(op(i,%)*10^(i-1),i=1..nops(%)) ;
            if issqr(%) then
                return true;
            end if;
        end do:
        for p in {0,1,4,5,6,9} do
            [p,op(ndgs)] ;
            add(op(i,%)*10^(i-1),i=1..nops(%)) ;
            if issqr(%) then
                return true;
            end if;
        end do:
        return false;
    end if;
end proc:
n := 1;
c := 1;
while n <= 10000 do
    if isA224955(c) then
        printf("%d %d\n",n,c) ;
        n := n+1 ;
    end if;
    c := c+1 ;
end do: # R. J. Mathar, Mar 14 2016

Mathematica

Module[{nn=300,pre=Range[9],app={0,1,4,5,6,9}},Select[Range[nn],(!IntegerQ[ Sqrt[ #]]) && (AnyTrue[Sqrt[pre*10^IntegerLength[#]+#],IntegerQ] || AnyTrue[ Sqrt[ 10#+app],IntegerQ])&]] (* Harvey P. Dale, Feb 27 2022 *)