This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A068133 #10 Sep 21 2024 19:19:28 %S A068133 0,1,3,6,28,78,1596,5995,67896,887778,15997996,398988876,9876799878, %T A068133 299789989975,35998897988976,589598998999878,78999997699698778, %U A068133 7987899888859999878,1998997979958978979995,539799799988999999688778 %N A068133 First triangular number with digit sum = n-th triangular number. %C A068133 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. %C A068133 From _Jon E. Schoenfield_, Jun 29 2010: (Start) %C A068133 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.) %C A068133 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) %F A068133 a(n) = A000217(A068134(n)). - _Andrew Howroyd_, Sep 21 2024 %Y A068133 Cf. A068127, A068128, A068129, A068130, A068131, A068132, A068134. %K A068133 base,nonn %O A068133 0,3 %A A068133 _Amarnath Murthy_, Feb 21 2002 %E A068133 More terms from Larry Reeves (larryr(AT)acm.org), Jun 17 2002 %E A068133 Term a(0) inserted and terms a(18) and a(19) added by _Jon E. Schoenfield_, Jun 29 2010