cp's OEIS Frontend

A380428 Numbers k for which nonnegative integers x and y exist such that k is the concatenation of x and y as well as k = (x + y)^2.

%I A380428 #20 Apr 27 2025 15:04:21
%S A380428 81,100,2025,3025,88209,494209,4941729,7441984,24502500,25502500,
%T A380428 52881984,60481729,300814336,493817284,6049417284,6832014336,
%U A380428 20408122449,21948126201,33058148761,35010152100,43470165025,101558217124,108878221089,123448227904,127194229449,152344237969
%N A380428 Numbers k for which nonnegative integers x and y exist such that k is the concatenation of x and y as well as k = (x + y)^2.
%C A380428 Subsequence of A000290.
%C A380428 From _David A. Corneth_, Apr 26 2025: (Start)
%C A380428 If y has q digits then a term m is of the form (x + y) = 10^q * x + y. Choosing some y we can solve for x (the equation is a quadratic with respect to x) and see if it produces a term.
%C A380428 y comes from A238712.
%C A380428 The sequence is infinite; it contains (25*100^i +- 5*10^i)^2 = concat(25*100^i +- 5*10^i, 25*100^i) for all i >= 0.
%C A380428 Neither x nor y can have a leading 0. (End)
%H A380428 David A. Corneth, <a href="/A380428/b380428.txt">Table of n, a(n) for n = 1..96</a> (terms <= 10^20)
%e A380428 2025 is in the sequence because (20 + 25)^2 = 2025.
%e A380428 100 is in the sequence because (10 + 0)^2 = 100.
%e A380428 88209 is in the sequence because (88 + 209)^2 = 88209.
%e A380428 From _David A. Corneth_, Apr 26 2025: (Start)
%e A380428 9801 is not in the sequence even though (98 + 01)^2 = 9801 but 01 has a leading 0 which is disallowed.
%e A380428 If a term m ends in y = 209 where y has three digits we have 10^3*x + y = (x + y)^2. Solving for x gives x = 88 or x = 494 corresponding to terms 88209 and 494209. (End)
%p A380428 A380428:=proc(n)
%p A380428     option remember;
%p A380428     local a,i,k,x,y;
%p A380428     if n=1 then
%p A380428         81
%p A380428     elif n=2 then
%p A380428         100
%p A380428     else
%p A380428         for a from isqrt(procname(n-1))+1 do
%p A380428             k:=length(a^2);
%p A380428             for i to k-1 do
%p A380428                 x:=floor(a^2/10^i);
%p A380428                 y:=a^2-x*10^i;
%p A380428                 if x+y=a and length(x)+length(y)=k then
%p A380428                     return a^2
%p A380428                 fi
%p A380428             od
%p A380428         od
%p A380428     fi;
%p A380428 end proc;
%p A380428 seq(A380428(n),n=1..26);
%Y A380428 Cf. A000290, A238712.
%Y A380428 Cf. A115527, A115528, A115529, A115530, A115531, A115532, A115533, A115534, A115535, A115536, A115537, A115538, A115539, A115540, A115541, A115542, A115543, A115544, A115545, A115546, A115547, A115548, A115549, A115550, A115551, A115552, A115553, A115554, A115555, A115556.
%K A380428 nonn,base,easy
%O A380428 1,1
%A A380428 _Felix Huber_, Jan 25 2025