A224935 Numbers n such that a positive number m <= n exists such that n-m, n+m, and n*m are triangular numbers.

19, 171, 271, 1428, 2178, 3781, 4053, 9303, 19459, 37980, 51238, 52669, 71316, 97083, 127014, 147978, 188074, 411675, 733591, 1018171, 1620010, 1701078, 1753416, 2159496, 2642781, 4542678, 5244753, 7337736, 10217611, 12251265, 16212831, 17100132, 35839545, 43400544

Offset

1,1

Comments

Sequence of corresponding m's begins: 9, 105, 135, 525, 2100, 1890, 225, 3417, 14994, 23445, 15192, 26334, 19635, 40467, 1764, 142725, 171054, 382755, 366795, 998865, 8100, ...

Conjectures:

1. The sequence is infinite.

2. There is only one m for each n (this is true for n < 2^26).

Links

Table of n, a(n) for n=1..34.

Example

The following three are triangular numbers:
271-135 = 136,
271+135 = 406,
271*135 = 36585.
So 271 is in the sequence.

Mathematica

pnmQ[n_]:=Length[Select[Table[{n-m, n+m, n m}, {m, n}], AllTrue[ Sqrt[ 8#+1], OddQ]&]]>0; Select[Range[20000], pnmQ] (* The program generates the first nine terms of the sequence. To generate more, increase the Range constant, but the program may take a long time to run. *) (* Requires Mathematica version 10 or later *) (* Harvey P. Dale, Jun 08 2021 *)

Prog

(Java)
public class A224935 {
  public static long sr0 = 1;
  public static boolean[] isTriang = new boolean[5 << 25];
  public static boolean isTriangular(long a) {
    long b, s, sr = sr0;
    while (a < sr*(sr+1)/2)  sr>>=1;
    b = sr>>1;
    while (b!=0) {
        s = sr+b;
        if (a >= s*(s+1)/2)  sr = s;
        b>>=1;
    }
    if (a == sr*(sr+1)/2)  return true;
    return false;
  }
  public static void main (String[] args) {
      for (int i=0; i*(i+1) < 5 << 26; ++i) isTriang[i*(i+1)/2] = true;
      for (long a = 0; a < 5 << 24; ++a) {
        long s = 1L << 30, tn = 0, count = 0, lastTn = 0;
        while (a*a < s*(s+1)/2)  s>>=1;
        sr0 = s;
        for (long i = 1; tn < a; ++i) {
          long b = a - tn;
          if (isTriang[(int)(a*2-tn)])
            if (isTriangular(a*(a-tn)))  { ++count; lastTn = tn; }
          tn += i;
        }
        if (count>0) System.out.printf("\n%d %d  %d ", a, a-lastTn, count);
        if ((a & 0x3fff)==0)  System.out.printf(".");
      }
  }
}

Crossrefs

Cf. A000217, A224954.

Sequence in context: A158966 A038864 A036155 * A047644 A010935 A282288

Adjacent sequences: A224932 A224933 A224934 * A224936 A224937 A224938

Keyword

nonn

Author

Alex Ratushnyak, Apr 20 2013

Status

approved