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A224935 Numbers n such that a positive number m <= n exists such that n-m, n+m, and n*m are triangular numbers. 2
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 (list; graph; refs; listen; history; text; internal format)
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.

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

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Last modified July 17 08:40 EDT 2019. Contains 325097 sequences. (Running on oeis4.)