A224242 Numbers k such that k^2 XOR (k+1)^2 is a square, and k^2 XOR (k-1)^2 is a square, where XOR is the bitwise logical XOR operator.

0, 4, 24, 44, 112, 480, 1984, 8064, 32512, 130560, 263160, 278828, 340028, 523264, 2095104, 8384512, 25239472, 32490836, 33546240, 134201344, 536838144, 2147418112

Offset: 1

Author

Alex Ratushnyak, Apr 01 2013

Keywords

• nonn
• base
• less

Comments

A subsequence of A221643: k's such that A221643(k) = A221643(k-1) + 1.

A059153 is a subsequence. Terms that are not in A059153: 0, 44, 263160, 278828, 340028, 25239472, 32490836. Conjecture: the subsequence of non-A059153 terms is infinite.

Crossrefs

Cf. A221643, A059153.

Programs

C

#include 
#include 
int main() {
  unsigned long long a, i, t;
  for (i=0; i < (1L<<32)-1; ++i) {
      a = (i*i) ^ ((i+1)*(i+1));
      t = sqrt(a);
      if (a != t*t) continue;
      a = (i*i) ^ ((i-1)*(i-1));
      t = sqrt(a);
      if (a != t*t) continue;
      printf("%llu, ", i);
  }
  return 0;
}

Mathematica

Select[Range[0,84*10^5],AllTrue[{Sqrt[BitXor[#^2,(#+1)^2]],Sqrt[BitXor[#^2,(#-1)^2] ]},IntegerQ]&] (* The program generates the first 16 terms of the sequence. *) (* Harvey P. Dale, Nov 10 2022 *)