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A068133 First triangular number with digit sum = n-th triangular number. 2
0, 1, 3, 6, 28, 78, 1596, 5995, 67896, 887778, 15997996, 398988876, 9876799878, 299789989975, 35998897988976, 589598998999878, 78999997699698778, 7987899888859999878, 1998997979958978979995, 539799799988999999688778 (list; graph; refs; listen; history; text; internal format)



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)


Table of n, a(n) for n=0..19.


Cf. A068127, A068128, A068129, A068130, A068131, A068132, A068133.

Sequence in context: A102428 A128056 A068808 * A220823 A024497 A007228

Adjacent sequences:  A068130 A068131 A068132 * A068134 A068135 A068136




Amarnath Murthy, Feb 21 2002


More terms from Larry Reeves (larryr(AT)acm.org), Jun 17 2002

Term a(0) inserted and terms a(18) and a(19) added by Jon E. Schoenfield, Jun 29 2010



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Last modified January 26 14:08 EST 2020. Contains 331280 sequences. (Running on oeis4.)