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

1,1

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

Links

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

Example

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

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

Prog

(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.')
(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

Crossrefs

Cf. A167035.

Sequence in context: A297760 A167035 A167042 * A164763 A164752 A151651

Adjacent sequences: A242265 A242266 A242267 * A242269 A242270 A242271

Keyword

nonn,base

Author

Reiner Moewald, Aug 16 2014

Status

approved