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

225, 5625, 5405625, 23765625, 2127515625, 58503515625, 51921031640625, 250727431640625, 20090404775390625, 608180644775390625, 498431438615478515625, 2642208974615478515625, 189450791534674072265625, 6319494849134674072265625, 9981411957966851806640625

Offset: 1

Author

Reiner Moewald, Aug 16 2014

Keywords

• nonn
• base

Comments

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)).

Subsequence of A167035. - Michel Marcus, Sep 08 2014

Examples

			225 = 15*15 and 1225 = 35*35.
		

Links

• Reiner Moewald, Table of n, a(n) for n = 1..102

Crossrefs

Cf. A167035.

Programs

Maple

A:= {}:
for m from 3 to 100 do
  cand1:= floor(log[5](1/2*(1+sqrt(2))*10^(m/2)));
  cand2:= floor(log[5](2*(1+sqrt(2))*(5/2)^(m/2)));
  s1:= 5^cand1 - 10^m/4/5^cand1;
  s2:=  2^m/4*5^cand2 - 5^(m-cand2);
  if s1^2 >= 10^(m-1) then A:= A union {s1^2} fi;
  if s2^2 >= 10^(m-1) then A:= A union {s2^2} fi;
od:
A; # Robert Israel, Sep 08 2014

PARI

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

Python

import math
def power(a, n):
   pow = 1
   for i in range(0, n):
      pow = pow * a
   return pow
end = 50
for n in range(1, end):
   l1 = 1/math.log(5)*(math.log(math.sqrt(2)-1)+(n-2)/2*math.log(2))+ n/2
   u1 = 1/math.log(5)*(math.log(math.sqrt(11)-1)+(n-3)/2*math.log(2))+ (n-1)/2
   if math.ceil(l1) == math.floor(u1) and math.ceil(l1)>0:
      p = math.ceil(l1)
      x = power(5, p)*(-1)+power(2, n-2)*power(5, n-p)
      print(x*x)
   l2 = 1/math.log(5)*(math.log(math.sqrt(11)+1)+(n-3)/2*math.log(2))+ (n-1)/2
   u2 = 1/math.log(5)*(math.log(math.sqrt(2)+1)+(n-2)/2*math.log(2))+ n/2
   if math.ceil(l2) == math.floor(u2) and math.ceil(l2)>0:
      p = math.ceil(l2)
      x = power(5, p)-power(2, n-2)*power(5, n-p)
      print(x*x)
print('End.')