A220752 Terms of A220698 that appear in A224218.

3854, 3854, 3035, 3035, 3035, 3035, 3854, 4644, 4644, 4644, 4644, 4644, 3854, 15846, 4644, 4644, 4644, 4644, 4644, 22918, 15846, 15846, 10225, 10225, 10225, 10225, 15846, 22918, 15846, 13364, 13364, 13364, 13364, 10225, 10225, 10225, 10225, 15846, 13364, 13364, 22918, 45012

Offset: 1

Author

Alex Ratushnyak, Apr 13 2013

Keywords

• nonn
• base

Comments

Terms of A220698 excluding terms that do not appear in A224218.

Indices of XOR-positive triangular numbers such that the generated triangular number is also XOR-positive (definition: triangular(i) is XOR-positive if triangular(i) XOR triangular(i+1) = triangular(k) for some k). XOR is the bitwise logical exclusive-or operator.

Conjecture: the sequence is infinite.

The subsequence with only odd terms begins: 3035, 3035, 3035, 3035, 10225, 10225, 10225, 10225, 10225, 10225, 10225, 10225, 171449, 171449, 236985, 171449, 339249.

Crossrefs

Cf. A000217, A224218, A220698.

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, n, r, t;
  for (n=0; n < (1L<<32)-1; n++) {
    a = (n*(n+1)/2) ^ ((n+1)*(n+2)/2);
    t = rootTriangular(a);
    if (a == t*(t+1)/2) {
        a ^= (t+1)*(t+2)/2;
        r = rootTriangular(a);
        if (a == r*(r+1)/2)  printf("%llu, ", t);
    }
  }
}