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A257649 Squares that are the concatenation of two integers (without leading zeros) the sum of which is also a square.

%I A257649 #44 Jun 08 2024 15:44:16
%S A257649 36,81,169,196,324,361,576,729,841,1156,1521,1681,1764,2809,3249,3481,
%T A257649 4356,5625,6084,6241,6724,7396,7569,7744,7921,8281,9216,12321,12544,
%U A257649 12769,12996,13689,15129,16384,17424,18769,19881,24964,25600,31684,32041,34596,36864,38416,39601
%N A257649 Squares that are the concatenation of two integers (without leading zeros) the sum of which is also a square.
%C A257649 Squares that can be split up in more than one way, e.g., 729 (72 + 9 and 7 + 29), appear only once.
%C A257649 The number of such squares is infinite, since 39...960...01 (the numbers of the digits 9 and 0 is equal) can be split up into 3 and 9...960..01 with 3 + 9...960...01 = (100...0-2)^2 and 39...960...01 = (2*100...0 - 1)^2.
%C A257649 From _Robert G. Wilson v_, Aug 06 2015: (Start)
%C A257649 Number of terms < 10^k: 0, 2, 9, 27, 66, 149, 370, 910, 2164, 5325, 12916, 29448, ..., .
%C A257649 Terms which are members of A257649 in more than one way: 729, 7569, 15129, 56169, 86436, 123201, ..., .
%C A257649 Terms which are members of A257649 in more than two way: 881377344, 3784833441, 39999600001, 54444755556, 71111288889, 89999400001, 159999200001, 321111488889, 751111688889, ..., .
%C A257649 Least term which is a member of A257649 in k ways: 36, 729, 881377344, 399999960000001, ..., . (End)
%H A257649 Robert G. Wilson v, <a href="/A257649/b257649.txt">Table of n, a(n) for n = 1..10000</a> (first 909 terms from Reiner Moewald)
%e A257649 36 = 6^2 and 3 + 6 = 9 = 3^2.
%t A257649 f[n_] := Block[{a, b, c, k = 1, idn = IntegerDigits@ n, lng, lst = {}}, lng = Length@ idn; While[k < lng, a = FromDigits[ Take[idn, {1, k}]]; b = FromDigits[ Take[idn, {k + 1, lng}]]; c = a*10^(lng - k) + b; If[b > 0 && Floor[1 + Log10@ b] == lng - k && IntegerQ@ Sqrt[a + b], AppendTo[lst, c]]; k++]; Length@ lst]; k = 1; lst = {}; While[k < 201, If[ f[k^2] > 0, AppendTo[lst, k^2]]; k++]; lst (* _Robert G. Wilson v_, Aug 06 2015 *)
%t A257649 ctiQ[n_]:=AnyTrue[Total/@Select[Table[FromDigits/@TakeDrop[IntegerDigits[n],d],{d,IntegerLength[ n]-1}],IntegerLength[#[[1]]]+IntegerLength[#[[2]]] ==IntegerLength[ n]&],IntegerQ[ Sqrt[#]]&]; Select[Range[200]^2,ctiQ] (* _Harvey P. Dale_, Jun 04 2023 *)
%o A257649 (Python)
%o A257649 import math
%o A257649 print("Start")
%o A257649 list =[]
%o A257649 for i in range(1,1000):
%o A257649    a = i*i
%o A257649    b = str(a)
%o A257649    l = len(b)
%o A257649    for j in range(1, l):
%o A257649       a_1 = b[:j]
%o A257649       a_2 = b[j:]
%o A257649       c = int(a_1)+int(a_2)
%o A257649       sqrt_c = int(math.sqrt(int(c)))
%o A257649       if (sqrt_c * sqrt_c == c) and (int(a_2[:1]) > 0):
%o A257649          if not a in list:
%o A257649             list.append(a)
%o A257649 print(list)
%o A257649 print("End")
%Y A257649 Subsequence of A052041.
%K A257649 nonn,base
%O A257649 1,1
%A A257649 _Reiner Moewald_, Jul 25 2015