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 A224241 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. 1
 0, 3, 130456, 342096, 1226720, 291575011, 379894587, 523040160, 15216609776, 136622606520 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,2 COMMENTS A subsequence of A221643. Conjecture: the sequence is infinite. LINKS PROG (C) #include #include int main() {   unsigned long long a, i, t;   for (i=0; i < (1L<<32)-2; ++i) {       a = (i*i) ^ ((i+1)*(i+1));       t = sqrt(a);       if (a != t*t) continue;       a = (i*i) ^ ((i+2)*(i+2));       t = sqrt(a);       if (a != t*t) continue;       printf("%llu, ", i);   }   return 0; } (Java) class A224241 {         static public BigInteger isqrt(final BigInteger n)         {                 if ( n.compareTo(BigInteger.ZERO) < 0 )                         throw new ArithmeticException("Negative argument "+ n.toString()) ;                 BigInteger x  ;                 final int bl = n.bitLength() ;                 if ( bl > 120)                         x = n.shiftRight(bl/2-1) ;                 else                 {                         final double resul= Math.sqrt(n.doubleValue()) ;                         x = new BigInteger(""+Math.round(resul)) ;                 }                 final BigInteger two = new BigInteger("2") ;                 while ( true)                 {                         BigInteger x2 = x.pow(2) ;                         BigInteger xplus2 = x.add(BigInteger.ONE).pow(2) ;                         if ( x2.compareTo(n) <= 0 && xplus2.compareTo(n) > 0)                                 return x ;                         xplus2 = xplus2.subtract(x.shiftLeft(2)) ;                         if ( xplus2.compareTo(n) <= 0 && x2.compareTo(n) > 0)                                 return x.subtract(BigInteger.ONE) ;                         xplus2 = x2.subtract(n).divide(x).divide(two) ;                         x = x.subtract(xplus2) ;                 }         }     static public void main(String[] argv)     {         for(BigInteger k = BigInteger.ZERO ;  ; k= k.add(BigInteger.ONE) )         {             final BigInteger k2 = k.pow(2) ;             final BigInteger kplus1 = k.add(BigInteger.ONE) ;             final BigInteger k12 = kplus1.pow(2) ;             final BigInteger xor1 = k2.xor(k12) ;             final BigInteger roo1 = isqrt(xor1) ;             if ( roo1.pow(2).compareTo(xor1) == 0 )             {                 final BigInteger k22 = kplus1.add(BigInteger.ONE).pow(2) ;                 final BigInteger xor2 = k2.xor(k22) ;                 final BigInteger roo2 = isqrt(xor2) ;                 if ( roo2.pow(2).compareTo(xor2) == 0 )                     System.out.println(k) ;             }         }     } } // R. J. Mathar, Apr 25 2013 CROSSREFS Cf. A221643. Sequence in context: A086785 A159577 A116536 * A178505 A306594 A003544 Adjacent sequences:  A224238 A224239 A224240 * A224242 A224243 A224244 KEYWORD nonn,base,more,less AUTHOR Alex Ratushnyak, Apr 01 2013 STATUS approved

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Last modified May 28 15:04 EDT 2022. Contains 354115 sequences. (Running on oeis4.)