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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 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

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Last modified June 23 05:01 EDT 2021. Contains 345395 sequences. (Running on oeis4.)