A375824 Triangular numbers whose sum of digits is 9.

36, 45, 153, 171, 351, 630, 1035, 1431, 2016, 3240, 3321, 4005, 8001, 10440, 13041, 13203, 16110, 21321, 23220, 25200, 101025, 105111, 114003, 222111, 320400, 321201, 1010331, 1241100, 1313010, 1400301, 2013021, 2031120, 2410110, 4020030, 10006101, 11203011, 20012301, 32004000, 32012001, 33020001

Offset: 1

Author

Robert Israel, Aug 30 2024

Keywords

• nonn
• base

Comments

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.

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

Examples

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

Links

• Robert Israel, Table of n, a(n) for n = 1..95

Crossrefs

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

Programs

Maple

F:= proc(d,s) option remember;
# d-digit numbers with sum of digits s
      local R,i;
      R:= {};
      for i from 0 to min(s,9) do
        R:= R union map(t -> 10*t+i, procname(d-1,s-i))
      od;
      R
end proc:
F(1,0):= {}:
for i from 1 to 9 do F(1,i):= {i} od:
sort(convert(`union`(seq(select(t -> issqr(1+8*t), F(d,9)),d=1..12)),list));

Mathematica

Select[Range[10000](Range[10000]+1)/2,DigitSum[#]==9 &] (* Stefano Spezia, Sep 01 2024 *)

PARI

select(x->(sumdigits(x)==9), vector(10000, n, n*(n+1)/2)) \\ Michel Marcus, Aug 31 2024