A226470 -id:A226470 - cp's OEIS frontend

A226471 Numbers n such that n^2 XOR triangular(n) is a triangular number. XOR is the bitwise logical exclusive-or operator.

0, 1, 3, 7, 15, 25, 31, 63, 113, 127, 189, 200, 255, 381, 481, 499, 511, 765, 1004, 1011, 1023, 1533, 1785, 1808, 1985, 2023, 2035, 2047, 3069, 3199, 3255, 3577, 3810, 4071, 4083, 4095, 4446, 6141, 6399, 7161, 8065, 8135, 8167, 8179, 8191, 12285, 12799, 14279, 14280

Offset: 1

Alex Ratushnyak, Jun 08 2013

Indices of triangular numbers in A226470. Numbers n such that A226470(n) and A226470(n+1) are triangular numbers: 0, 14279, 491279, 16251935, 29358023, 528478271, ...

Cf. A000217, A000290, A226470, A224218.

Select[Range[0,15000],OddQ[Sqrt[8*BitXor[#^2,(#(#+1))/2]+1]]&] (* Harvey P. Dale, Jul 22 2024 *)

import math
for n in range(100000000):
  a = (n*n) ^ (n*(n+1)//2)
  r = int(math.sqrt(a*2))
  if r*(r+1)==a*2: print(n, end=', ')

A226472 Numbers n such that n^2 XOR triangular(n) is a perfect square. XOR is the bitwise logical exclusive-or operator.

Original entry on oeis.org

0, 1, 6, 8, 4086, 24136, 162297, 7868054, 29792904, 22666693375

Offset: 1

Alex Ratushnyak, Jun 08 2013

Indices of perfect squares in A226470. No other terms below 2^35. Roots of generated squares: 0, 0, 7, 10, 2915, 29506, 149434, 6328037, 27602118, 20243443647.

			8^2 XOR triangular(8) = 64 XOR 36 = 100, because 100 is a perfect square, 8 is in the sequence.

Cf. A000217, A000290, A226470, A221643.

#include 
#include 
int main() {
  for (unsigned long long a, r, n=0; n < (1ULL<<32); ++n) {
    a = (n*n) ^ (n*(n+1)/2);
    r = sqrt(a);
    if (r*r==a)  printf("%llu, ", n);
  }
  return 0;
}

cp's OEIS Frontend

A226471 Numbers n such that n^2 XOR triangular(n) is a triangular number. XOR is the bitwise logical exclusive-or operator.

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