A115556
Numbers whose square is the concatenation of two numbers 9*m and m.
Original entry on oeis.org
12857142857142857142857142857142857143, 25714285714285714285714285714285714286, 117391304347826086956521739130434782608695652173913043478261
Offset: 1
-
F:= proc(d) local R,F,t,b,r,q,s,m0,x0,k;
R:= NULL;
F:= ifactors(9*10^d+1)[2];
b:= mul(t[1]^floor(t[2]/2),t=F);
for r in numtheory:-divisors(b) do
x0:= (9*10^d+1)/r;
m0:= x0/r;
for k from ceil(sqrt(10^(d-1)/m0)) to floor(sqrt(10^d/m0)) do
R:= R, x0*k;
od
od;
R
end proc:
sort(map(F, [$1..90])); # Robert Israel, Aug 24 2023
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.
Original entry on oeis.org
81, 100, 2025, 3025, 88209, 494209, 4941729, 7441984, 24502500, 25502500, 52881984, 60481729, 300814336, 493817284, 6049417284, 6832014336, 20408122449, 21948126201, 33058148761, 35010152100, 43470165025, 101558217124, 108878221089, 123448227904, 127194229449, 152344237969
Offset: 1
2025 is in the sequence because (20 + 25)^2 = 2025.
100 is in the sequence because (10 + 0)^2 = 100.
88209 is in the sequence because (88 + 209)^2 = 88209.
From _David A. Corneth_, Apr 26 2025: (Start)
9801 is not in the sequence even though (98 + 01)^2 = 9801 but 01 has a leading 0 which is disallowed.
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)
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.
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A380428:=proc(n)
option remember;
local a,i,k,x,y;
if n=1 then
81
elif n=2 then
100
else
for a from isqrt(procname(n-1))+1 do
k:=length(a^2);
for i to k-1 do
x:=floor(a^2/10^i);
y:=a^2-x*10^i;
if x+y=a and length(x)+length(y)=k then
return a^2
fi
od
od
fi;
end proc;
seq(A380428(n),n=1..26);
Showing 1-2 of 2 results.
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