A375824 - OEIS

%I #11 Sep 01 2024 18:41:01

%S 36,45,153,171,351,630,1035,1431,2016,3240,3321,4005,8001,10440,13041,

%T 13203,16110,21321,23220,25200,101025,105111,114003,222111,320400,

%U 321201,1010331,1241100,1313010,1400301,2013021,2031120,2410110,4020030,10006101,11203011,20012301,32004000,32012001,33020001

%N Triangular numbers whose sum of digits is 9.

%C Infinite subsequences include 2 * 10^(2*k) + 13 * 10^k + 21, 2 * 10^(2*k) + 31 * 10^k + 120, 32 * 10^(2*k) + 4 * 10^k, and 32 * 10^(2*k) + 12 * 10^k + 1.

%C Conjecture: the last term not of one of those subsequences is a(53) =  210010000005.

%H Robert Israel, <a href="/A375824/b375824.txt">Table of n, a(n) for n = 1..95</a>

%e a(4) = 153 is a term because 153 = 17 * 18/2 is a triangular number and 1 + 5 + 3 = 9.

%p F:= proc(d,s) option remember;

%p # d-digit numbers with sum of digits s

%p       local R,i;

%p       R:= {};

%p       for i from 0 to min(s,9) do

%p         R:= R union map(t -> 10*t+i, procname(d-1,s-i))

%p       od;

%p       R

%p end proc:

%p F(1,0):= {}:

%p for i from 1 to 9 do F(1,i):= {i} od:

%p sort(convert(`union`(seq(select(t -> issqr(1+8*t), F(d,9)),d=1..12)),list));

%t Select[Range[10000](Range[10000]+1)/2,DigitSum[#]==9 &] (* _Stefano Spezia_, Sep 01 2024 *)

%o (PARI) select(x->(sumdigits(x)==9), vector(10000, n, n*(n+1)/2)) \\ _Michel Marcus_, Aug 31 2024

%Y Intersection of A000217 and A052223.  Contained in A117404 and A076713.

%K nonn,base

%O 1,1

%A _Robert Israel_, Aug 30 2024