A004157 -id:A004157 - cp's OEIS frontend

A068127 Triangular numbers with sum of digits = 3.

3, 21, 120, 210, 300, 10011, 20100, 2001000, 200010000, 20000100000, 2000001000000, 200000010000000, 20000000100000000, 2000000001000000000, 200000000010000000000, 20000000000100000000000, 2000000000001000000000000, 200000000000010000000000000, 20000000000000100000000000000

Offset: 1

Amarnath Murthy, Feb 21 2002

The sequence is unbounded, as the (2*10^k)-th triangular number is a term.

Michael S. Branicky, Table of n, a(n) for n = 1..504

Cf. A000217, A004157, A052217.

t = {}; Do[tri = n*(n+1)/2; If[Total[IntegerDigits[tri, 10]] == 3, AppendTo[t, tri]], {n, 1000000}]; t (* T. D. Noe, Jun 05 2012 *)
Select[Accumulate[Range[2*10^6]],Total[IntegerDigits[#]]==3&] (* Harvey P. Dale, Jun 22 2021 *)
Sort @ Select[Plus @@@ (10^Select[Tuples[Range[0, 29], 3], Min[Differences[#]] >= 0 &]), IntegerQ[Sqrt[8*# + 1]] &] (* Amiram Eldar, May 19 2022 *)

from math import isqrt
from itertools import count, islice
def istri(n): return (lambda x: x == isqrt(x)**2)(8*n+1)
def agen(): yield from filter(istri, (10**i + 10**j + 10**k for i in count(0) for j in range(i+1) for k in range(j+1)))
print(list(islice(agen(), 20))) # Michael S. Branicky, May 14 2022

More terms from Sascha Kurz, Mar 06 2002
One additional term (a(12)) from Harvey P. Dale, May 14 2022
More terms and offset changed to 1 from Michael S. Branicky, May 14 2022

A145389 Digital roots of triangular numbers.

Original entry on oeis.org

0, 1, 3, 6, 1, 6, 3, 1, 9, 9, 1, 3, 6, 1, 6, 3, 1, 9, 9, 1, 3, 6, 1, 6, 3, 1, 9, 9, 1, 3, 6, 1, 6, 3, 1, 9, 9, 1, 3, 6, 1, 6, 3, 1, 9, 9, 1, 3, 6, 1, 6, 3, 1, 9, 9, 1, 3, 6, 1, 6, 3, 1, 9, 9, 1, 3, 6, 1, 6, 3, 1, 9, 9, 1, 3, 6, 1, 6, 3, 1, 9, 9, 1, 3, 6, 1, 6

Offset: 0

Reinhard Zumkeller, Oct 10 2008

Decimal expansion of 45387733/3333333330. - Enrique Pérez Herrero, Nov 14 2021

Index entries for linear recurrences with constant coefficients, signature (0,0,0,0,0,0,0,0,1).

Cf. A000217, A004157, A010888.

digitalRoot[n_Integer?Positive] := FixedPoint[Plus@@IntegerDigits[#]&,n]; Table[If[n==0,0,digitalRoot[n(n+1)/2]], {n,0,100}] (* Vladimir Joseph Stephan Orlovsky, May 02 2011 *)

a(n)=if(n, n=n*(n+1)/2%9; if(n, n, 9), 0) \\ Charles R Greathouse IV, Dec 19 2016

def A145389(n): return (9, 1, 3, 6, 1, 6, 3, 1, 9)[n%9] if n else 0 # Chai Wah Wu, Feb 09 2023

a(n) = A010888(A000217(n)).
Periodic sequence for n>0: a(n+9) = a(n);
a(A016777(n)) = 1; a(A007494(n)) <> 1;
a(A090570(n)) = A010888(A090570(n)).
a(n) = 1 + ((n^2 + n - 2)/2) mod 9. - Ant King, Apr 25 2009
G.f.: x(1 + 3x + 6x^2 + x^3 + 6x^4 + 3x^5 + x^6 + 9x^7 + 9x^8)/((1-x)(1 + x + x^2)(1 + x^3 + x^6)). - Ant King, Nov 16 2010

A050493 a(n) = sum of binary digits of n-th triangular number.

Original entry on oeis.org

0, 1, 2, 2, 2, 4, 3, 3, 2, 4, 5, 2, 4, 5, 4, 4, 2, 4, 5, 6, 4, 6, 7, 3, 4, 4, 7, 6, 5, 6, 5, 5, 2, 4, 5, 6, 5, 8, 6, 4, 5, 7, 6, 6, 8, 4, 5, 4, 4, 5, 8, 6, 5, 7, 7, 3, 6, 7, 8, 7, 6, 7, 6, 6, 2, 4, 5, 6, 5, 8, 7, 8, 4, 6, 8, 5, 8, 9, 4, 5, 5, 8, 7, 8, 8, 7, 8, 8, 7, 8, 12, 5, 6, 5, 6, 5, 4, 5, 8, 7, 8

Offset: 0

Klaus Strassburger (strass(AT)ddfi.uni-duesseldorf.de), Dec 27 1999

See A211201 for smallest numbers m such that a(m) = n. - Reinhard Zumkeller, Feb 04 2013

Cf. A000120, A000217, A004157.

a050493 = a000120 . a000217  -- Reinhard Zumkeller, Feb 04 2013

f[n_]:=Plus@@IntegerDigits[n,2]; lst={};Do[t=n*(n+1)/2;AppendTo[lst,f[t]],{n,6!}];lst (* Vladimir Joseph Stephan Orlovsky, Oct 10 2009 *)
Total[IntegerDigits[#,2]]&/@Accumulate[Range[0,100]] (* Harvey P. Dale, Jan 22 2012 *)

a(n)=hammingweight(n*(n+1)) \\ Charles R Greathouse IV, Nov 10 2015

a(n) = Sum_{i=1..floor(log_b(c(n)))+1} (floor(c(n)/b^(i-1)) - floor(c(n)/b^i)*b), b=2, n >= 1, a(0)=0, c(n)=A000217(n).
a(n) = A000120(A000217(n)). - Reinhard Zumkeller, Feb 04 2013
a(n) = [x^(n*(n+1)/2)] (1/(1 - x))*Sum_{k>=0} x^(2^k)/(1 + x^(2^k)). - Ilya Gutkovskiy, Mar 27 2018

A117511 Triangular numbers for which the sum of the digits equals the sum of the digits of the next triangular number.

Original entry on oeis.org

36, 153, 2556, 3240, 4851, 5778, 9045, 11628, 13041, 14535, 17766, 19503, 33930, 41328, 46665, 49455, 52326, 71253, 74691, 81810, 85491, 93096, 109278, 122265, 131328, 140715, 145530, 160461, 170820, 181503, 186966, 192510, 203841, 252405, 258840, 265356

Offset: 1

Luc Stevens (lms022(AT)yahoo.com), Apr 26 2006

Each term is divisible by 9.

			153 is in the sequence because (1) 153 is triangular number a(18), triangular number a(19)=171 and (2) 1+5+3=1+7+1.

Harvey P. Dale, Table of n, a(n) for n = 1..1000

Cf. A000217, A004157.

Transpose[With[{c=Partition[Accumulate[Range[2000]],2,1]}, Select[c, Total[IntegerDigits[First[#]]]==Total[IntegerDigits[Last[#]]]&]]] [[1]] (* Harvey P. Dale, Oct 18 2011 *)
(#(#+1))/2&/@(SequencePosition[Total[IntegerDigits[#]]&/@Accumulate[ Range[ 1000]],{x_,x_}][[All,1]]) (* Harvey P. Dale, Mar 02 2022 *)

s(a(n)) = s(a(n+1)), where s(n) is the sum of the digits of n.

Corrected by Harvey P. Dale, Oct 18 2011

A383356 a(n) = index of the smallest nonagonal number having the same digital sum as the n-th triangular number.

Original entry on oeis.org

1, 6, 3, 1, 3, 6, 4, 2, 2, 4, 5, 12, 4, 3, 6, 4, 2, 2, 4, 6, 3, 4, 12, 6, 4, 2, 11, 4, 5, 12, 13, 12, 5, 13, 2, 11, 4, 5, 12, 4, 12, 5, 13, 11, 2, 4, 5, 12, 4, 12, 5, 13, 2, 11, 4, 23, 12, 4, 12, 5, 13, 11, 2, 4, 5, 3, 13, 12, 5, 13, 11, 11, 4, 23, 12, 13, 3, 5, 4, 2, 2, 4

Offset: 1

Claude H. R. Dequatre, Apr 24 2025

From Robert Israel, Apr 24 2025: (Start)
If n == 0 or 8 (mod 9) then a(n) == 0 or 2 (mod 9).
If n == 1, 4 or 7 (mod 9) then a(n) == 1, 4 or 7 (mod 9).
If n == 2 or 6 (mod 9) then a(n) == 5 or 6 (mod 9).
If n == 3 or 5 (mod 9) then a(n) == 3 or 8 (mod 9). (End)

			For n = 2, the 2nd triangular number is (2^2+2)/2 = 3, its digital sum is 3 and the smallest nonagonal number having 3 as digital sum is (7*6^2 - 5*6)/2 = 111 whose index is 6, so a(2) = 6.
For n = 16, the 16-th triangular number is (16^2 +16)/2 = 136, its digital sum is 10 and the smallest nonagonal number having 10 as digital sum is (7*4^2 -5*4)/2 = 46 whose index is 4, so a(16) = 4.

Michel Lagneau, Table of n, a(n) for n = 1..10000

Cf. A000217, A001106, A007953, A004157.

ds:= n -> convert(convert(n,base,10),`+`):
v:= 0: R:= NULL:
for k from 1 to 200 do
   r:= ds(k*(k+1)/2);
   if assigned(W[r]) then R:= R,W[r]
   else do
     v:= v+1;
     s:= ds(v*(7*v-5)/2);
     if not assigned(W[s]) then W[s]:= v fi;
     if s = r then R:= R,v; break fi;
     od fi od:
R; # Robert Israel, Apr 24 2025

a(n) = my(s=sumdigits((n^2+n)/2)); k=1; while(sumdigits((7*k^2-5*k)/2)!=s, k++); k;

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A068127 Triangular numbers with sum of digits = 3.

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A145389 Digital roots of triangular numbers.

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A050493 a(n) = sum of binary digits of n-th triangular number.

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A117511 Triangular numbers for which the sum of the digits equals the sum of the digits of the next triangular number.

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A383356 a(n) = index of the smallest nonagonal number having the same digital sum as the n-th triangular number.

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