A115545
Numbers k such that the concatenation of 6*k with k gives a square.
Original entry on oeis.org
122449, 489796, 122448979591836734693877551020408163265306122449, 489795918367346938775510204081632653061224489796
Offset: 1
734694_122449 = 857143^2.
A115546
Numbers k such that k^2 is the concatenation of two numbers 6*m and m.
Original entry on oeis.org
857143, 1714286, 857142857142857142857142857142857142857142857143, 1714285714285714285714285714285714285714285714286
Offset: 1
857143^2 = 734694_122449.
A115547
Numbers k such that the concatenation of 7*k with k gives a square.
Original entry on oeis.org
11909262759924385633270321361058601134215500945179584121, 21172022684310018903591682419659735349716446124763705104, 33081285444234404536862003780718336483931947069943289225, 47637051039697542533081285444234404536862003780718336484
Offset: 1
A115548
Numbers k such that k^2 is the concatenation of two numbers m and 7*m.
Original entry on oeis.org
91304347826086956521739130434782608695652173913043478261, 121739130434782608695652173913043478260869565217391304348, 152173913043478260869565217391304347826086956521739130435, 182608695652173913043478260869565217391304347826086956522
Offset: 1
A115550
Numbers k such that k^2 is the concatenation of two numbers m and 8*m.
Original entry on oeis.org
18, 36, 168, 252, 336, 1668, 3336, 16668, 33336, 166668, 333336, 1666668, 3333336, 16666668, 22222224, 27777780, 33333336, 113149848, 114678900, 116207952, 117737004, 119266056, 120795108, 122324160, 123853212
Offset: 1
A115551
Numbers k such that the concatenation of 8*k with k gives a square.
Original entry on oeis.org
1, 4, 9, 89, 889, 8889, 88889, 888889, 8888889, 88888889, 888888889, 1026765584, 1066636289, 1107266436, 1148656025, 1190805056, 1233713529, 1277381444, 1321808801, 1366995600, 1412941841, 1459647524, 1507112649, 1555337216
Offset: 1
A115552
Numbers k such that k^2 is the concatenation of two numbers 8*m and m.
Original entry on oeis.org
9, 18, 27, 267, 2667, 26667, 266667, 2666667, 26666667, 266666667, 2666666667, 9063180828, 9237472767, 9411764706, 9586056645, 9760348584, 9934640523, 10108932462, 10283224401, 10457516340, 10631808279, 10806100218
Offset: 1
A115553
Numbers k such that the concatenation of k with 9*k gives a square.
Original entry on oeis.org
2041, 8164, 2366863905325444, 5325443786982249, 9467455621301776, 2040816326530612244897959183673469387755102041, 8163265306122448979591836734693877551020408164
Offset: 1
A115554
Numbers k such that k^2 is the concatenation of two numbers m and 9*m.
Original entry on oeis.org
14287, 28574, 15384615384615386, 23076923076923079, 30769230769230772, 14285714285714285714285714285714285714285714287, 28571428571428571428571428571428571428571428574
Offset: 1
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.
-
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);
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