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A258060
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Squares, without multiplicity, that are the concatenation of two integers (without leading zeros) the product of which is also a square.
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1
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49, 169, 361, 1225, 1444, 1681, 3249, 4225, 4900, 15625, 16900, 36100, 42025, 49729, 64009, 81225, 93025, 122500, 142129, 144400, 168100, 225625, 237169, 324900, 414736, 422500, 490000, 519841, 819025, 950625, 970225, 1024144, 1442401, 1562500, 1600225, 1690000, 1692601, 2079364, 2304324
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OFFSET
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1,1
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COMMENTS
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Squares that can be split up in more than one way, e.g., 4950625 with sqrt(4 * 950625) = 1950 and sqrt(49 * 50625) = 1575, appear only once.
Squares that are members of this sequence in more than one way: 4950625, 495062500, 49506250000, 4950625000000, ..., . - Robert G. Wilson v, Aug 14 2015
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LINKS
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EXAMPLE
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169 = 13^2 can be split up into 16 and 9 and 16*9 = 144, a square.
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MAPLE
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p:= proc(k, n) local t; t:= n mod 10^k; t >= 10^(k-1) and issqr(t*(n-t)/10^k) end proc:
filter:= n -> ormap(p, [$1..ilog10(n)], n):
select(filter, [seq(i^2, i=1..10^4)]); # Robert Israel, Sep 22 2015
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MATHEMATICA
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f[n_] := Block[{idn = IntegerDigits@ n, c = 0, k = 1, lmt = Floor[1 + Log10@ n]}, While[k < lmt, m = Mod[n, 10^(lmt - k)]; If[ IntegerQ@ Sqrt[ FromDigits[ Take[idn, {1, k}]] m] && m > 0 && IntegerDigits[m] == Take[idn, {k + 1, -1}], c++]; k++]; c]; Select[ Range[1700]^2, f@# > 0 &] (* Robert G. Wilson v, Aug 13 2015 *)
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PROG
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(Python)
import math
list =[]
for i in range(1, 100000):
...a = i*i
...b = str(a)
...l = len(b)
...for j in range(1, l):
......a_1 = b[:j]
......a_2 = b[j:]
......c = int(a_1)* int(a_2)
......sqrt_c = int(math.sqrt(int(c)))
......if (sqrt_c * sqrt_c == c) and (int(a_2[:1]) > 0):
.........if not a in list:
............list.append(a)
.........list.append(a)
print(list)
(PARI) isok(n) = {if (issquare(n), len = #Str(n); for (k=1, len-1, na = n\10^k; nb = n%10^k; if (na && nb && (eval(Str(na, nb))==n) && issquare(na*nb), return (1)); ); ); } \\ Michel Marcus, Oct 09 2015
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CROSSREFS
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KEYWORD
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nonn,base
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AUTHOR
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STATUS
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approved
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