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A242268 Squares not ending in 00 that remain squares if prefixed with the digit 1.

%I A242268 #31 Feb 03 2025 18:57:55
%S A242268 225,5625,5405625,23765625,2127515625,58503515625,51921031640625,
%T A242268 250727431640625,20090404775390625,608180644775390625,
%U A242268 498431438615478515625,2642208974615478515625,189450791534674072265625,6319494849134674072265625,9981411957966851806640625
%N A242268 Squares not ending in 00 that remain squares if prefixed with the digit 1.
%C A242268 It can easily be shown that all squares that remain squares if prefixed with the digit 1 end in 00 or 25 and, moreover, that all squares ending in 00 are multiples of the squares ending in 5 (factor: 10^(2*n)).
%C A242268 Subsequence of A167035. - _Michel Marcus_, Sep 08 2014
%H A242268 Reiner Moewald, <a href="/A242268/b242268.txt">Table of n, a(n) for n = 1..102</a>
%e A242268 225 = 15*15 and 1225 = 35*35.
%p A242268 A:= {}:
%p A242268 for m from 3 to 100 do
%p A242268   cand1:= floor(log[5](1/2*(1+sqrt(2))*10^(m/2)));
%p A242268   cand2:= floor(log[5](2*(1+sqrt(2))*(5/2)^(m/2)));
%p A242268   s1:= 5^cand1 - 10^m/4/5^cand1;
%p A242268   s2:=  2^m/4*5^cand2 - 5^(m-cand2);
%p A242268   if s1^2 >= 10^(m-1) then A:= A union {s1^2} fi;
%p A242268   if s2^2 >= 10^(m-1) then A:= A union {s2^2} fi;
%p A242268 od:
%p A242268 A; # _Robert Israel_, Sep 08 2014
%o A242268 (Python)
%o A242268 import math
%o A242268 def power(a, n):
%o A242268    pow = 1
%o A242268    for i in range(0, n):
%o A242268       pow = pow * a
%o A242268    return pow
%o A242268 end = 50
%o A242268 for n in range(1, end):
%o A242268    l1 = 1/math.log(5)*(math.log(math.sqrt(2)-1)+(n-2)/2*math.log(2))+ n/2
%o A242268    u1 = 1/math.log(5)*(math.log(math.sqrt(11)-1)+(n-3)/2*math.log(2))+ (n-1)/2
%o A242268    if math.ceil(l1) == math.floor(u1) and math.ceil(l1)>0:
%o A242268       p = math.ceil(l1)
%o A242268       x = power(5, p)*(-1)+power(2, n-2)*power(5, n-p)
%o A242268       print(x*x)
%o A242268    l2 = 1/math.log(5)*(math.log(math.sqrt(11)+1)+(n-3)/2*math.log(2))+ (n-1)/2
%o A242268    u2 = 1/math.log(5)*(math.log(math.sqrt(2)+1)+(n-2)/2*math.log(2))+ n/2
%o A242268    if math.ceil(l2) == math.floor(u2) and math.ceil(l2)>0:
%o A242268       p = math.ceil(l2)
%o A242268       x = power(5, p)-power(2, n-2)*power(5, n-p)
%o A242268       print(x*x)
%o A242268 print('End.')
%o A242268 (PARI)
%o A242268 for(n=1,10^20,p=n^2;if(p%100,s=concat("1",Str(p));if(issquare(eval(s)),print1(p,", ")))) \\ _Derek Orr_, Aug 23 2014
%Y A242268 Cf. A167035.
%K A242268 nonn,base
%O A242268 1,1
%A A242268 _Reiner Moewald_, Aug 16 2014