A076713 Harshad (Niven) triangular numbers: triangular numbers which are divisible by the sum of their digits.

1, 3, 6, 10, 21, 36, 45, 120, 153, 171, 190, 210, 300, 351, 378, 465, 630, 666, 780, 820, 990, 1035, 1128, 1275, 1431, 1540, 1596, 1770, 2016, 2080, 2556, 2628, 2850, 2926, 3160, 3240, 3321, 3486, 3570, 4005, 4465, 4560, 4950, 5050, 5460, 5565, 5778, 5886

Offset: 1

Author

Shyam Sunder Gupta, Oct 26 2002

Keywords

• nonn
• base

Comments

Intersection of A000217 and A005349. - K. D. Bajpai, Aug 13 2014

Examples

			a(5)=21: 21 is a triangular number and also a Harshad number as 21 is divisible by 2+1=3. So 21 is Harshad triangular number.
		

Links

• K. D. Bajpai, Table of n, a(n) for n = 1..15436
• Shyam Sunder Gupta Fascinating Triangular Numbers

Crossrefs

Cf. A000217, A005349. Includes A037156(n) for n >= 2. Includes A068127.

Programs

Mathematica

TriangularNumberQ[k_] := If[IntegerQ[1/2 (Sqrt[1 + 8 k] - 1)], True, False]; Harshad[k_] := Select[Range[k], IntegerQ[ #/(Plus @@ IntegerDigits[ # ])] &]; TriangularHarshad[k_] := Select[Harshad[k], TriangularNumberQ[#] &]; TriangularHarshad[5886] (* Ant King, Dec 13 2010 *)
A076713 = {}; Do[k = n*(n + 1)*1/2; If[IntegerQ[k/(Plus @@ IntegerDigits[k])], AppendTo[A076713, k]], {n,1000}]; A076713  (* K. D. Bajpai, Aug 13 2014 *)