A261127 Triangular numbers t such that (sum of digits of t) + (product of digits of t) is a triangular number.

0, 3, 10, 105, 120, 136, 190, 210, 300, 406, 703, 780, 820, 1081, 1128, 1431, 1540, 1653, 1770, 1891, 1953, 2080, 2211, 2628, 2701, 2850, 3003, 3160, 3403, 3570, 4560, 4656, 5050, 5460, 7021, 7260, 7503, 8646, 8911, 9453, 10011, 10153, 11026, 12403, 14028, 15400

Offset

1,2

Comments

All the terms in this sequence are triangular, and hence 0 or 1 (mod 3).

Links

K. D. Bajpai, Table of n, a(n) for n = 1..10000

Example

a(6) = 136 = 16 * (16+1) / 2, that is triangular number. (1+3+6) + (1*3*6) = 28, which is 7th triangular number.
a(15) = 1128 = 47 * (47+1) / 2, that is triangular number. (1+1+2+8) + (1*1*2*8) = 28, which is 7th triangular number.

Maple

with(numtheory): A261127:= proc() local a, k, t; t:=n*(n+1)/2; a:= (add(d, d=convert(t, base, 10)) + mul(d, d=convert(t, base, 10))); k:=(-1 + sqrt(8*a + 1))/2; if k=floor(k) then RETURN (t); fi; end: seq(A261127 (), n=0..300);

Mathematica

A261127 = {}; Do[t = n*(n + 1)/2; k = Plus @@ IntegerDigits[t] + Times @@ IntegerDigits[t]; If[IntegerQ[( -1 + Sqrt[8*k + 1])/2], AppendTo[A261127, t]], {n, 0, 1000}]; A261127

Prog

(PARI) for(n =0, 500, t = n*(n+1)/2; k = (sumdigits(t)); d = digits (t); p = prod(i = 1, #d, d[i]); s = k+p; if(ispolygonal(s, 3), print1(t, ", ")));
(Magma) [n*(n+1) div 2: n in [0..100] | IsSquare(8*k+1) where k is (&+Intseq(n*(n+1) div 2) + &*Intseq(n*(n+1) div 2))];

Crossrefs

Cf. A000217, A007953, A007954, A061762, A214629.

Sequence in context: A203903 A173325 A233257 * A083108 A117664 A091342

Adjacent sequences: A261124 A261125 A261126 * A261128 A261129 A261130

Keyword

nonn,base

Author

K. D. Bajpai, Aug 09 2015

Status

approved