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

1,2

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.

Links

Table of n, a(n) for n=1..22.

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 *)

Prog

(C)
#include <stdio.h>
#include <math.h>
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;
}

Crossrefs

Cf. A221643, A059153.

Sequence in context: A031117 A174178 A139245 * A066770 A080380 A364278

Adjacent sequences: A224239 A224240 A224241 * A224243 A224244 A224245

Keyword

nonn,base,less

Author

Alex Ratushnyak, Apr 01 2013

Status

approved