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

1,1

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.

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.

Example

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.

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 *)

Crossrefs

Cf. A023110, A000290.

Sequence in context: A213857 A011864 A045919 * A006856 A249013 A211964

Adjacent sequences: A224952 A224953 A224954 * A224956 A224957 A224958

Keyword

nonn,base

Author

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

Status

approved