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

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

Offset: 1

Author

Reiner Moewald, Jul 26 2015

Keywords

• nonn
• base

Comments

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

Examples

			169 = 13^2 can be split up into 16 and 9 and 16*9 = 144, a square.
		

Links

• Robert G. Wilson v, Table of n, a(n) for n = 1..1386 (first 200 terms from Reiner Moewald)

Crossrefs

Subsequence of A039686.

Programs

Maple

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

Mathematica

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 *)

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

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)