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A242268 Squares not ending in 00 that remain squares if prefixed with the digit 1. 4
225, 5625, 5405625, 23765625, 2127515625, 58503515625, 51921031640625, 250727431640625, 20090404775390625, 608180644775390625, 498431438615478515625, 2642208974615478515625, 189450791534674072265625, 6319494849134674072265625, 9981411957966851806640625 (list; graph; refs; listen; history; text; internal format)
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

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Last modified May 26 06:59 EDT 2022. Contains 354074 sequences. (Running on oeis4.)