cp's OEIS Frontend

1. The sequence is unbounded, as the (2*10^k +3)-th triangular number is a term. 2. The sum of the digits of triangular numbers in most cases is a triangular number. 3. Conjecture: For every triangular number T there exist infinitely many triangular numbers with sum of digits = T.

1. The sequence is unbounded, as the (5*10^k + 6)-th triangular number is a term. 2. The sum of the digits of triangular numbers in most cases is a triangular number. 3. Conjecture: For every triangular number T there exist infinitely many triangular numbers with sum of digits = T.

The sum of the digits of triangular numbers in most cases is a triangular number. Conjecture: For every triangular number T there exist infinitely many triangular numbers with sum of digits = T.

From Jon E. Schoenfield, Jun 29 2010: (Start)

For any positive k < 132, it is true that more than half of the positive triangular numbers from T(1) through T(k) have a triangular digit sum. However, for any k > 132, more than half of the positive triangular numbers from T(1) through T(k) have a nontriangular digit sum. (At k = 132, there are 66 triangular and 66 nontriangular.)

There exist only finitely many triangular numbers whose digit sum is T(0)=0 or T(1)=1: T(0)=0 is, of course, the only one with digit sum 0, and T(1)=1 and T(4)=10 are the only two with digit sum 1. However, for digit sums equal to each of at least the next several triangular numbers, the conjecture can be easily confirmed by observing that, e.g., T(2), T(20), T(200), T(2000), etc., all have digit sum T(2)=3; T(2+1), T(20+1), T(200+1), T(2000+1), etc., all have digit sum T(3)=6; T(20+2), T(200+2), T(2000+2), T(20000+2), etc., all have digit sum T(4)=10; and, similarly, for all sufficiently large values of j, triangular numbers of the form T(2*10^j+m), where m = 3, 9, 23, 34, 132, 368, 1332, 3943, 19388, 88248, 244948, 1788848, 9838483, 19994343, respectively, will have digit sums T(5)=15, T(6)=21, ..., T(18)=171, respectively. (End)

The sequence is unbounded, as the (10^k + 9)-th triangular number for k>1 is a term.

The sum of the digits of triangular numbers in most cases is a triangular number.

Conjecture: For every triangular number T there exist infinitely many triangular numbers with sum of digits = T.

Conjecture: No entry is zero.

Next term (A002473(n) = 21) is either 0 or > 10^100. - David Wasserman, Jan 27 2005

a(17) = 0, because A002473(17) = 21 = 3*7 and the last two digits of every member of A002473 greater than 7 contain a digit other than 1, 3, and 7. - Robert Israel, Nov 23 2019

Similarly, a(n) = 0 whenever n >=17 and A002473(n) is coprime to 10. - Robert Israel, Nov 24 2019