A224241 - OEIS

%I #13 Jul 19 2016 11:36:09

%S 0,3,130456,342096,1226720,291575011,379894587,523040160,15216609776,

%T 136622606520

%N Numbers k such that k^2 XOR (k+1)^2 is a square, and k^2 XOR (k+2)^2 is also a square, where XOR is the bitwise logical exclusive-or operator.

%C A subsequence of A221643.

%C Conjecture: the sequence is infinite.

%o (C)

%o #include <stdio.h>

%o #include <math.h>

%o int main() {

%o   unsigned long long a, i, t;

%o   for (i=0; i < (1L<<32)-2; ++i) {

%o       a = (i*i) ^ ((i+1)*(i+1));

%o       t = sqrt(a);

%o       if (a != t*t) continue;

%o       a = (i*i) ^ ((i+2)*(i+2));

%o       t = sqrt(a);

%o       if (a != t*t) continue;

%o       printf("%llu, ", i);

%o   }

%o   return 0;

%o }

%o (Java)

%o class A224241 {

%o         static public BigInteger isqrt(final BigInteger n)

%o         {

%o                 if ( n.compareTo(BigInteger.ZERO) < 0 )

%o                         throw new ArithmeticException("Negative argument "+ n.toString()) ;

%o                 BigInteger x  ;

%o                 final int bl = n.bitLength() ;

%o                 if ( bl > 120)

%o                         x = n.shiftRight(bl/2-1) ;

%o                 else

%o                 {

%o                         final double resul= Math.sqrt(n.doubleValue()) ;

%o                         x = new BigInteger(""+Math.round(resul)) ;

%o                 }

%o                 final BigInteger two = new BigInteger("2") ;

%o                 while ( true)

%o                 {

%o                         BigInteger x2 = x.pow(2) ;

%o                         BigInteger xplus2 = x.add(BigInteger.ONE).pow(2) ;

%o                         if ( x2.compareTo(n) <= 0 && xplus2.compareTo(n) > 0)

%o                                 return x ;

%o                         xplus2 = xplus2.subtract(x.shiftLeft(2)) ;

%o                         if ( xplus2.compareTo(n) <= 0 && x2.compareTo(n) > 0)

%o                                 return x.subtract(BigInteger.ONE) ;

%o                         xplus2 = x2.subtract(n).divide(x).divide(two) ;

%o                         x = x.subtract(xplus2) ;

%o                 }

%o         }

%o     static public void main(String[] argv)

%o     {

%o         for(BigInteger k = BigInteger.ZERO ;  ; k= k.add(BigInteger.ONE) )

%o         {

%o             final BigInteger k2 = k.pow(2) ;

%o             final BigInteger kplus1 = k.add(BigInteger.ONE) ;

%o             final BigInteger k12 = kplus1.pow(2) ;

%o             final BigInteger xor1 = k2.xor(k12) ;

%o             final BigInteger roo1 = isqrt(xor1) ;

%o             if ( roo1.pow(2).compareTo(xor1) == 0 )

%o             {

%o                 final BigInteger k22 = kplus1.add(BigInteger.ONE).pow(2) ;

%o                 final BigInteger xor2 = k2.xor(k22) ;

%o                 final BigInteger roo2 = isqrt(xor2) ;

%o                 if ( roo2.pow(2).compareTo(xor2) == 0 )

%o                     System.out.println(k) ;

%o             }

%o         }

%o     }

%o }

%o // _R. J. Mathar_, Apr 25 2013

%Y Cf. A221643.

%K nonn,base,more,less

%O 1,2

%A _Alex Ratushnyak_, Apr 01 2013