cp's OEIS Frontend

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

%I A224955 #22 Feb 27 2022 11:11:35
%S A224955 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 A224955 67,69,72,76,78,84,89,90,96,102,108,115,116,122,129,136,152,156,160,
%U A224955 168,176,184,193,202,209,211,216,220,230,240,241,249,250,260,270,280
%N A224955 Numbers that are not squares, but can become squares by prepending or appending one additional digit.
%C A224955 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 A224955 Christian N. K. Anderson, <a href="/A224955/b224955.txt">Table of n, a(n) for n = 1..10000</a>
%H A224955 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 A224955 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 A224955 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 A224955 isA224955 := proc(n)
%p A224955     local p,ndgs;
%p A224955     if issqr(n) then
%p A224955         return false;
%p A224955     else
%p A224955         ndgs := convert(n,base,10) ;
%p A224955         for p from 1 to 9 do
%p A224955             [op(ndgs),p] ;
%p A224955             add(op(i,%)*10^(i-1),i=1..nops(%)) ;
%p A224955             if issqr(%) then
%p A224955                 return true;
%p A224955             end if;
%p A224955         end do:
%p A224955         for p in {0,1,4,5,6,9} do
%p A224955             [p,op(ndgs)] ;
%p A224955             add(op(i,%)*10^(i-1),i=1..nops(%)) ;
%p A224955             if issqr(%) then
%p A224955                 return true;
%p A224955             end if;
%p A224955         end do:
%p A224955         return false;
%p A224955     end if;
%p A224955 end proc:
%p A224955 n := 1;
%p A224955 c := 1;
%p A224955 while n <= 10000 do
%p A224955     if isA224955(c) then
%p A224955         printf("%d %d\n",n,c) ;
%p A224955         n := n+1 ;
%p A224955     end if;
%p A224955     c := c+1 ;
%p A224955 end do: # _R. J. Mathar_, Mar 14 2016
%t A224955 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 A224955 Cf. A023110, A000290.
%K A224955 nonn,base
%O A224955 1,1
%A A224955 _Kevin L. Schwartz_ and _Christian N. K. Anderson_, Apr 21 2013