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A258060 Squares, without multiplicity, that are the concatenation of two integers (without leading zeros) the product of which is also a square.

%I A258060 #47 May 22 2025 10:21:43
%S A258060 49,169,361,1225,1444,1681,3249,4225,4900,15625,16900,36100,42025,
%T A258060 49729,64009,81225,93025,122500,142129,144400,168100,225625,237169,
%U A258060 324900,414736,422500,490000,519841,819025,950625,970225,1024144,1442401,1562500,1600225,1690000,1692601,2079364,2304324
%N A258060 Squares, without multiplicity, that are the concatenation of two integers (without leading zeros) the product of which is also a square.
%C A258060 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.
%C A258060 Squares that are members of this sequence in more than one way: 4950625, 495062500, 49506250000, 4950625000000, ..., . - _Robert G. Wilson v_, Aug 14 2015
%H A258060 Robert G. Wilson v, <a href="/A258060/b258060.txt">Table of n, a(n) for n = 1..1386</a> (first 200 terms from Reiner Moewald)
%e A258060 169 = 13^2 can be split up into 16 and 9 and 16*9 = 144, a square.
%p A258060 p:= proc(k,n) local t; t:= n mod 10^k; t >= 10^(k-1) and issqr(t*(n-t)/10^k) end proc:
%p A258060 filter:= n -> ormap(p, [$1..ilog10(n)], n):
%p A258060 select(filter, [seq(i^2, i=1..10^4)]); # _Robert Israel_, Sep 22 2015
%t A258060 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 *)
%o A258060 (Python)
%o A258060 import math
%o A258060 list =[]
%o A258060 for i in range(1,100000):
%o A258060    a = i*i
%o A258060    b = str(a)
%o A258060    l = len(b)
%o A258060    for j in range(1, l):
%o A258060       a_1 = b[:j]
%o A258060       a_2 = b[j:]
%o A258060       c = int(a_1)* int(a_2)
%o A258060       sqrt_c = int(math.sqrt(int(c)))
%o A258060       if (sqrt_c * sqrt_c == c) and (int(a_2[:1]) > 0):
%o A258060          if not a in list:
%o A258060             list.append(a)
%o A258060          list.append(a)
%o A258060 print(list)
%o A258060 (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
%Y A258060 Subsequence of A039686.
%K A258060 nonn,base
%O A258060 1,1
%A A258060 _Reiner Moewald_, Jul 26 2015