A224955 - OEIS

%I #22 Feb 27 2022 11:11:35

%S 2,3,5,6,8,10,12,14,19,21,22,24,28,29,32,40,41,44,48,52,56,57,61,62,

%T 67,69,72,76,78,84,89,90,96,102,108,115,116,122,129,136,152,156,160,

%U 168,176,184,193,202,209,211,216,220,230,240,241,249,250,260,270,280

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

%C 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.

%H Christian N. K. Anderson, <a href="/A224955/b224955.txt">Table of n, a(n) for n = 1..10000</a>

%H Christian N. K. Anderson, <a href="/A224955/a224955.txt">List of squares</a> that can be formed by concatenating one digit to the first 10000 terms.

%H Christian N. K. Anderson, <a href="/A224955/a224955.gif">Ulam spiral</a> of a(n), with brighter colors corresponding to the number of ways a term may become a square.

%e 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.

%p isA224955 := proc(n)

%p     local p,ndgs;

%p     if issqr(n) then

%p         return false;

%p     else

%p         ndgs := convert(n,base,10) ;

%p         for p from 1 to 9 do

%p             [op(ndgs),p] ;

%p             add(op(i,%)*10^(i-1),i=1..nops(%)) ;

%p             if issqr(%) then

%p                 return true;

%p             end if;

%p         end do:

%p         for p in {0,1,4,5,6,9} do

%p             [p,op(ndgs)] ;

%p             add(op(i,%)*10^(i-1),i=1..nops(%)) ;

%p             if issqr(%) then

%p                 return true;

%p             end if;

%p         end do:

%p         return false;

%p     end if;

%p end proc:

%p n := 1;

%p c := 1;

%p while n <= 10000 do

%p     if isA224955(c) then

%p         printf("%d %d\n",n,c) ;

%p         n := n+1 ;

%p     end if;

%p     c := c+1 ;

%p end do: # _R. J. Mathar_, Mar 14 2016

%t 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 *)

%Y Cf. A023110, A000290.

%K nonn,base

%O 1,1

%A _Kevin L. Schwartz_ and _Christian N. K. Anderson_, Apr 21 2013