A220518 Numbers n such that A193232(n) is a triangular number.

0, 1, 5, 9, 45, 49, 101, 153, 1569, 7163, 7171, 8162, 18974, 18976, 24467, 33490, 60290, 63046, 359539, 551494, 1769418, 2813691, 4140392, 4649729, 6675935, 9653486, 59131393, 158169499, 243345386, 588183781, 1315697727, 1387290631, 1522472645, 1879098546

Offset: 1

Author

Alex Ratushnyak, Mar 27 2013

Keywords

• base
• nonn

Comments

Numbers k such that bitwise XOR of first k triangular numbers is a triangular number.

Crossrefs

Cf. A193232.

Cf. A145827 (bitwise XOR of first k squares is a square).

Programs

C

#include 
typedef unsigned long long U64;
U64 rootTriangular(U64 a) {
    U64 sr = 1L<<32, s, b;
    if (a < (sr/2)*(sr+1)) {
      sr>>=1;
      while (a < sr*(sr+1)/2)  sr>>=1;
    }
    for (b = sr>>1; b; b>>=1) {
        s = sr+b;
        if (a >= s*(s+1)/2)  sr = s;
    }
    return sr;
}
int main() {
  U64 a=0, i, t;
  for (i=0; i < 1L<<32; ++i) {
      a ^= i*(i+1)/2;
      t = rootTriangular(a);
      if (a == t*(t+1)/2)  printf("%llu\n", i);
  }
  return 0;
}