A037156 -id:A037156 - cp's OEIS frontend

A062691 Triangular numbers that contain exactly 2 different digits.

10, 15, 21, 28, 36, 45, 78, 91, 171, 300, 595, 990, 1711, 2211, 3003, 5050, 5151, 5565, 5995, 6555, 8778, 10011, 66066, 222111, 255255, 333336, 500500, 600060, 828828, 887778, 1188111, 5656566, 22221111, 50005000, 51151555, 88877778, 2222211111, 5000050000

Offset: 1

Erich Friedman, Jul 04 2001

For n > 2, A309597(n) is a term. - Seiichi Manyama, Sep 15 2019

The other known infinite families of terms are A037156(n) for n > 1, A319170(n), and A383942(n). - David Radcliffe, Aug 25 2025

			300 is triangular and contains the digits 0 and 3.

David Radcliffe, Table of n, a(n) for n = 1..116 (terms 1..51 from Seiichi Manyama).

Cf. A000217, A045914 (all digits the same), A213516, A213518, A309597.

Select[Accumulate[Range[14000]],Count[DigitCount[#],Except[0]]==2&] (* Harvey P. Dale, Nov 27 2011 *)

for(k=0, 1e5, if(#Set(digits(j=k*(k+1)/2))==2, print1(j", "))) \\ Seiichi Manyama, Sep 15 2019

A038544 a(n) = Sum_{i=0..10^n} i^3.

Original entry on oeis.org

1, 3025, 25502500, 250500250000, 2500500025000000, 25000500002500000000, 250000500000250000000000, 2500000500000025000000000000, 25000000500000002500000000000000, 250000000500000000250000000000000000, 2500000000500000000025000000000000000000

Offset: 0

Marvin Ray Burns

These terms k = x.y satisfy Diophantine equation x.y = (x+y)^2, when x and y have the same number of digits, "." means concatenation, and y may not begin with 0. So, this is a subsequence of A350870 and A238237. - Bernard Schott, Jan 20 2022

			a(1) = Sum_{i=0..10} i^3 = (Sum_{i=0..10} i)^2 = 3025.

Cf. A037156, A238237, A350869, A350870.

sumcu(n) = for(x=0,n,y=10^x;z=y^2*(y+1)^2/4;(print1(z","))) \\ Cino Hilliard, Jun 18 2007

a(n) = (10^n+1)^2 * 10^(2*n) / 4.
From Bernard Schott, Jan 20 2022: (Start)
a(n) = A037156(n)^2.
a(n) = A350869(n) + 10^(3*n). (End)

Edited by N. J. A. Sloane, Jul 02 2008 at the suggestion of R. J. Mathar

A076713 Harshad (Niven) triangular numbers: triangular numbers which are divisible by the sum of their digits.

Original entry on oeis.org

1, 3, 6, 10, 21, 36, 45, 120, 153, 171, 190, 210, 300, 351, 378, 465, 630, 666, 780, 820, 990, 1035, 1128, 1275, 1431, 1540, 1596, 1770, 2016, 2080, 2556, 2628, 2850, 2926, 3160, 3240, 3321, 3486, 3570, 4005, 4465, 4560, 4950, 5050, 5460, 5565, 5778, 5886

Offset: 1

Shyam Sunder Gupta, Oct 26 2002

Intersection of A000217 and A005349. - K. D. Bajpai, Aug 13 2014

			a(5)=21: 21 is a triangular number and also a Harshad number as 21 is divisible by 2+1=3. So 21 is Harshad triangular number.

K. D. Bajpai, Table of n, a(n) for n = 1..15436
Shyam Sunder Gupta Fascinating Triangular Numbers

Cf. A000217, A005349. Includes A037156(n) for n >= 2. Includes A068127.

TriangularNumberQ[k_] := If[IntegerQ[1/2 (Sqrt[1 + 8 k] - 1)], True, False]; Harshad[k_] := Select[Range[k], IntegerQ[ #/(Plus @@ IntegerDigits[ # ])] &]; TriangularHarshad[k_] := Select[Harshad[k], TriangularNumberQ[#] &]; TriangularHarshad[5886] (* Ant King, Dec 13 2010 *)
A076713 = {}; Do[k = n*(n + 1)*1/2; If[IntegerQ[k/(Plus @@ IntegerDigits[k])], AppendTo[A076713, k]], {n,1000}]; A076713  (* K. D. Bajpai, Aug 13 2014 *)

A350869 a(n) = Sum_{i=0..10^n-1} i^3.

Original entry on oeis.org

0, 2025, 24502500, 249500250000, 2499500025000000, 24999500002500000000, 249999500000250000000000, 2499999500000025000000000000, 24999999500000002500000000000000, 249999999500000000250000000000000000, 2499999999500000000025000000000000000000

Offset: 0

Bernard Schott, Jan 20 2022

These terms k = x.y satisfy equation x.y = (x+y)^2, when x and y have the same number of digits, "." means concatenation, and y may not begin with 0. So, this is a subsequence of A350870 and A238237.

			a(1) = Sum_{i=0..9} i^3 = (Sum_{i=0..9} i)^2 = 2025.

Index entries for linear recurrences with constant coefficients, signature (11100,-11100000,1000000000).

Cf. A000217, A037156, A037182, A038544, A238237.

a[n_] := (10^n*(10^n - 1)/2)^2; Array[a, 11, 0] (* Amiram Eldar, Jan 20 2022 *)

a(n) = my(x=10^n-1); (x*(x+1)/2)^2; \\ Michel Marcus, Jan 22 2022

a(n) = 10^(2n) * (10^n-1)^2 / 4 = A037182(n)^2.
a(n) = A000217(10^n-1)^2.
a(n) = A038544(n) - 10^(3*n).

A067520 Triangular numbers whose index is a multiple of the sum of their digits.

Original entry on oeis.org

1, 10, 21, 45, 55, 120, 171, 300, 465, 666, 820, 1035, 1485, 1830, 2016, 2211, 2628, 2850, 2926, 3321, 4095, 5050, 5565, 5886, 7260, 8001, 8911, 10011, 10440, 13203, 14196, 16290, 17955, 18145, 18528, 19701, 20910, 22155, 23436, 24310, 29646

Offset: 1

Amarnath Murthy, Feb 14 2002

			T(30) = 465 and 30 = 2*(4+6+5).

Robert Israel, Table of n, a(n) for n = 1..10000

Includes A037156.

f:= proc(n) local t;
   t:= n*(n+1)/2;
   if n mod convert(convert(t,base,10),`+`) = 0 then return t fi
end proc:
map(f, [$1..300]); # Robert Israel, Jan 18 2024

PolygonalNumber[Select[Range[300], Divisible[#, Total[IntegerDigits[# (# + 1) / 2]]]&]] (* Paolo Xausa, Jan 18 2024 *)

More terms from Sascha Kurz, Mar 18 2002

Offset corrected by Sean A. Irvine, Dec 17 2023

A293686 8-digit numbers (padded with leading zeros where necessary) in which the sum of the number consisting of the first four digits and the number consisting of the last four digits equals the number consisting of the middle four digits.

Original entry on oeis.org

0, 10099, 10100, 20199, 20200, 30299, 30300, 40399, 40400, 50499, 50500, 60599, 60600, 70699, 70700, 80799, 80800, 90899, 90900, 100999, 101000, 111099, 111100, 121199, 121200, 131299, 131300, 141399, 141400, 151499, 151500, 161599, 161600, 171699, 171700

Offset: 1

Harvey P. Dale, Oct 14 2017

Zero can be a leading digit.
The sequence is the 8-digit analog to A263194.
This sequence contains 5050 terms. - David A. Corneth, Oct 14 2017

			131299 is a term because 0013 + 1299 = 1312 and 1312 is the string of the middle four digits of 00131299.

David A. Corneth, Table of n, a(n) for n = 1..5050 (full sequence; first 1045 terms from Vincenzo Librandi)

Cf. A037156, A263194.

[n: n in [0..2*10^5] | (n div 10000+n) mod 10000 eq (n div 100) mod 10000]; // Vincenzo Librandi, Oct 15 2017

dn8Q[n_]:=Module[{d=PadLeft[IntegerDigits[n],8,0]},FromDigits[ d[[1;;4]]]+ FromDigits[ d[[5;;8]]]==FromDigits[d[[3;;6]]]]; Select[Range[ 0,10^6], dn8Q]

is(n) = n < 10^8 && n\10000 + n%10000 == (n \ 100) % 10000 \\ David A. Corneth, Oct 14 2017

seq() = {my(t = 0, res = List(), c1, c2); while(t < 10^8, listput(res, t); c2 = (t\10000)%100; if(c2 < 99, t+= 10100, c1 = t\10^6; t = (c1+1)*10^6 + (c1 + 2)*10^4 + 98 - c1)); for(i=2, #res, if(res[i] > 10^6, listsort(res); return(res)); listput(res, res[i]-1))} \\ (this program produces the full sequence) David A. Corneth, Oct 16 2017

A383942 a(n) = (8*10^(2n) - 10^(n+1) + 2) / 9.

Original entry on oeis.org

78, 8778, 887778, 88877778, 8888777778, 888887777778, 88888877777778, 8888888777777778, 888888887777777778, 88888888877777777778, 8888888888777777777778, 888888888887777777777778, 88888888888877777777777778, 8888888888888777777777777778

Offset: 1

David Radcliffe, Aug 18 2025

This is one of four infinite families of triangular numbers consisting of two different digits. The other three families are A319170, A037156 (n>1), and A309597 (n>2).

Paolo Xausa, Table of n, a(n) for n = 1..495
Index entries for linear recurrences with constant coefficients, signature (111,-1110,1000).

Cf. A000217, A037156, A062691, A073551, A256340, A309597, A319170.

Subsequence of: A119216, A119230, A119236.

A383942[n_] := (8*10^(2*n) - 10^(n+1) + 2)/9; Array[A383942, 15] (* or *)
LinearRecurrence[{111, -1110, 1000}, {78, 8778, 887778}, 15] (* Paolo Xausa, Aug 27 2025 *)

def A383942(n): return (8*10**(2*n)-10**(n+1)+2)//9

a(n) = A000217(A073551(n+1)).

G.f.: 6*x*(13 + 20*x)/((1 - x)*(1 - 10*x)*(1 - 100*x)). - Stefano Spezia, Aug 19 2025

A352057 Triangular numbers whose nonzero digits are all the same.

Original entry on oeis.org

0, 1, 3, 6, 10, 55, 66, 300, 666, 990, 3003, 5050, 10011, 66066, 500500, 600060, 50005000, 5000050000, 500000500000, 50000005000000, 5000000050000000, 500000000500000000, 50000000005000000000, 5000000000050000000000, 500000000000500000000000, 50000000000005000000000000

Offset: 1

Steven Lu, Mar 02 2022

This sequence may correspond to "monochromatic step squads" in the British animation "Numberblocks".

Conjecture: the largest term in this sequence whose nonzero digits are not 5 is 600060.

Supersequence of A037156.
Cf. A000217, A118668, A125289, A343811.
Cf. A352148 (indices of these triangular numbers).

(* Method1 *)
NonZeroQ[n_Integer] := n != 0; Select[
Table[n (n + 1)/2, {n, 0, 1000000}],
Length[Tally[Select[IntegerDigits[#], NonZeroQ]]] == 1 &]
(* Method2 *)
Sort[Select[
  Flatten[Outer[Times,
    Table[FromDigits[IntegerDigits[n, 2]], {n, 2^16 - 1}], Range[9]]],
   IntegerQ[Sqrt[8 # + 1]] &]]

isok(k) = my(d=digits(k*(k+1)/2)); d = select(x->(x!=0), d); #Set(d)<=1;
lista(nn) = {for (n=0, nn, if (isok(n), print1(n*(n+1)/2, ", ")););} \\ Michel Marcus, Mar 02 2022

from sympy import integer_nthroot
from sympy.utilities.iterables import multiset_permutations
def istri(n): return integer_nthroot(8*n+1, 2)[1]
def zplus1(digits):
    if digits == 1: yield 0
    for d1 in "123456789":
        digset = "0"*(digits-1) + d1*(digits-1)
        for mp in multiset_permutations(digset, digits-1):
            t = int(d1 + "".join(mp))
            yield t
def afind(maxdigits):
    for digits in range(1, maxdigits+1):
        for t in zplus1(digits):
            if istri(t):
                print(t, end=", ")
afind(22) # Michael S. Branicky, Mar 02 2022

a(24)-a(25) from Michael S. Branicky, Mar 02 2022

cp's OEIS Frontend

A062691 Triangular numbers that contain exactly 2 different digits.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

A038544 a(n) = Sum_{i=0..10^n} i^3.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

PARI

Formula

Extensions

A076713 Harshad (Niven) triangular numbers: triangular numbers which are divisible by the sum of their digits.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

A350869 a(n) = Sum_{i=0..10^n-1} i^3.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A067520 Triangular numbers whose index is a multiple of the sum of their digits.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Maple

Mathematica

Extensions

A293686 8-digit numbers (padded with leading zeros where necessary) in which the sum of the number consisting of the first four digits and the number consisting of the last four digits equals the number consisting of the middle four digits.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Magma

Mathematica

PARI

PARI

A383942 a(n) = (8*10^(2n) - 10^(n+1) + 2) / 9.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

Python

Formula