A045914 -id:A045914 - 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

A118668 Number of distinct digits needed to write the n-th triangular number in decimal representation.

Original entry on oeis.org

1, 1, 1, 1, 2, 2, 2, 2, 2, 2, 1, 1, 2, 2, 3, 3, 3, 3, 2, 3, 3, 3, 3, 3, 2, 3, 3, 3, 3, 3, 3, 3, 3, 3, 2, 3, 1, 3, 3, 3, 3, 3, 3, 3, 2, 4, 3, 3, 3, 3, 4, 4, 4, 3, 4, 4, 4, 4, 2, 3, 4, 3, 4, 4, 3, 4, 2, 3, 4, 4, 4, 3, 3, 4, 3, 4, 3, 2, 4, 4, 4, 3, 3, 4, 4, 3, 4, 3, 4, 3, 4, 4, 4, 4, 3, 4, 3, 4, 4, 4, 2, 2, 3, 3, 4

Offset: 0

Reinhard Zumkeller, May 19 2006

0 < a(n) <= 10;

a(n) = A043537(A000217(n)).

			n=99: 99*(99+1)/2 = 4950 -> a(99) = #{0,4,5,9} = 4;
see A119033 for an overview of sequences with terms composed of not more than 3 distinct digits.
n=100: 100*(100+1)/2 = 5050 -> a(100) = #{0,5} = 2;

Reinhard Zumkeller, Table of n, a(n) for n = 0..10000

Cf. A000217, A043537, A045914, A213517, A213518, A255678.

a118668 = a043537 . a000217
a118668_list = map a043537 a000217_list
-- Reinhard Zumkeller, Jul 11 2015

Length[Union[IntegerDigits[#]]]&/@Accumulate[Range[0,110]] (* Harvey P. Dale, Jul 23 2012 *)

A213516 Triangular numbers having only 1 or 2 different digits in base 10.

Original entry on oeis.org

0, 1, 3, 6, 10, 15, 21, 28, 36, 45, 55, 66, 78, 91, 171, 300, 595, 666, 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

Offset: 1

T. D. Noe, Jun 21 2012

The list of triangular numbers containing only one digit (A045914) is finite. This list is infinite because numbers like 8888777778, 222222111111, and 500000500000 occur an infinite number of times.

A309597 is a subsequence. - Seiichi Manyama, Sep 14 2019

Seiichi Manyama, Table of n, a(n) for n = 1..55

Cf. A000217, A045914, A213517, A309597, A319170.

Cf. A119033 (has list of sequences related to digits in triangular numbers).

[n*(n+1)/2: n in [0..10^5] | #Set(Intseq(n*(n+1) div 2)) le 2]; // Bruno Berselli, Oct 27 2012

t = {}; Do[tri = n*(n+1)/2; If[Length[Union[IntegerDigits[tri]]] <= 2, AppendTo[t, tri]], {n, 0, 10^5}]; t
Select[Accumulate[Range[0,20000]],Count[DigitCount[#],0]>7&] (* Harvey P. Dale, Sep 03 2020 *)

A241787 Triangular numbers which have one or more occurrences of exactly four different digits.

Original entry on oeis.org

1035, 1275, 1326, 1378, 1485, 1540, 1596, 1653, 1830, 1953, 2016, 2145, 2346, 2415, 2485, 2701, 2850, 3081, 3160, 3240, 3486, 3570, 3741, 3916, 4095, 4186, 4278, 4371, 4560, 4753, 4851, 4950, 5460, 5671, 6105, 6328, 6903, 7021, 7140, 7260, 7381, 7503, 8256

Offset: 1

Colin Barker, Apr 28 2014

The first term having a repeated digit is 10153.

Colin Barker, Table of n, a(n) for n = 1..2000

Cf. A045914, A062691, A162304, A241788, A241789, A241790, A241791, A241792, A241812.

Cf. A000217.

Select[Accumulate[Range[200]],Count[DigitCount[#],0]==6&] (* Harvey P. Dale, Jun 29 2022 *)

s=[]; for(n=0, 300, if(#vecsort(eval(Vec(Str(n*(n+1)/2))), , 8)==4, s=concat(s, n*(n+1)/2))); s

A241788 Triangular numbers which have one or more occurrences of exactly five different digits.

Original entry on oeis.org

10296, 12403, 13695, 14028, 14365, 14706, 16290, 17205, 19306, 19503, 21736, 21945, 23871, 24310, 24531, 24753, 24976, 27495, 29403, 30628, 30876, 32640, 32896, 34716, 34980, 37128, 37401, 37950, 39621, 40186, 41328, 41905, 42195, 43071, 43956, 46971, 47586

Offset: 1

Colin Barker, Apr 28 2014

The first term having a repeated digit is 100576.

Colin Barker, Table of n, a(n) for n = 1..1000

Cf. A045914, A062691, A162304, A241787, A241789, A241790, A241791, A241792, A241812.

Cf. A000217.

s=[]; for(n=0, 500, if(#vecsort(eval(Vec(Str(n*(n+1)/2))), , 8)==5, s=concat(s, n*(n+1)/2))); s

A241789 Triangular numbers which have one or more occurrences of exactly six different digits.

Original entry on oeis.org

102378, 103285, 104653, 106953, 108345, 109278, 109746, 120786, 124750, 132870, 135460, 137026, 138075, 150426, 152076, 154290, 158203, 162735, 168490, 170236, 174936, 178503, 189420, 190653, 194376, 197506, 198765, 203841, 205761, 215496, 219453, 231540

Offset: 1

Colin Barker, Apr 28 2014

The first term having a repeated digit is 1004653.

Colin Barker, Table of n, a(n) for n = 1..2500

Cf. A045914, A062691, A162304, A241787, A241788, A241790, A241791, A241792, A241812.

Cf. A000217.

Select[Table[(n(n+1))/2,{n,447,681}],Length[Union[ IntegerDigits[ #]]] == 6&] (* Harvey P. Dale, Feb 17 2018 *)

s=[]; for(n=0, 800, if(#vecsort(eval(Vec(Str(n*(n+1)/2))), , 8)==6, s=concat(s, n*(n+1)/2))); s

A241790 Triangular numbers which have one or more occurrences of exactly seven different digits.

Original entry on oeis.org

1024596, 1047628, 1053426, 1069453, 1073845, 1078246, 1203576, 1234806, 1345620, 1360425, 1362075, 1386945, 1390278, 1405326, 1430586, 1439056, 1462905, 1486950, 1493856, 1547920, 1549680, 1590436, 1602945, 1624503, 1642578, 1679028, 1684530, 1693720

Offset: 1

Colin Barker, Apr 28 2014

The first term having a repeated digit is 10149765.

Colin Barker, Table of n, a(n) for n = 1..2000

Cf. A045914, A062691, A162304, A241787, A241788, A241789, A241791, A241792, A241812.

Cf. A000217.

s=[]; for(n=0, 2500, if(#vecsort(eval(Vec(Str(n*(n+1)/2))), , 8)==7, s=concat(s, n*(n+1)/2))); s

A241791 Triangular numbers which have one or more occurrences of exactly eight different digits.

Original entry on oeis.org

10348975, 10623745, 10725396, 10869453, 10934826, 12347965, 12357906, 12487503, 12647935, 12673095, 12784096, 13862745, 14756028, 14826735, 15237960, 15298746, 15304278, 15879430, 16247850, 16384950, 17084935, 17502486, 17543926, 17829406

Offset: 1

Colin Barker, Apr 28 2014

The first term having a repeated digit is 100642578.

Colin Barker, Table of n, a(n) for n = 1..2000

Cf. A045914, A062691, A162304, A241787, A241788, A241789, A241790, A241792, A241812.

Cf. A000217.

Select[Accumulate[Range[7000]],Length[Union[IntegerDigits[#]]]==8&] (* Harvey P. Dale, Feb 09 2019 *)

s=[]; for(n=0, 7000, if(#vecsort(eval(Vec(Str(n*(n+1)/2))), , 8)==8, s=concat(s, n*(n+1)/2))); s

A241792 Triangular numbers which have one or more occurrences of exactly nine different digits.

Original entry on oeis.org

102738945, 120784653, 120893475, 124875306, 126794850, 129854670, 137904528, 142087653, 143287056, 147069825, 149826705, 152783940, 153694278, 160249753, 162495378, 168370425, 173249805, 189725460, 192540876, 193405278, 197438256, 207193546, 230469715

Offset: 1

Colin Barker, Apr 28 2014

The first term having a repeated digit is 1004976528.

Colin Barker, Table of n, a(n) for n = 1..1500

Cf. A045914, A062691, A162304, A241787, A241788, A241789, A241790, A241791, A241812.

Cf. A000217.

s=[]; for(n=0, 40000, if(#vecsort(eval(Vec(Str(n*(n+1)/2))), , 8)==9, s=concat(s, n*(n+1)/2))); s

A241812 Triangular numbers which have one or more occurrences of exactly ten different digits.

Original entry on oeis.org

1062489753, 1239845706, 1256984730, 1520843976, 1539264870, 1597283460, 1684930275, 1952843760, 1957346028, 1978236450, 2197480365, 2367098415, 2418079653, 2503948761, 2634980715, 2718609453, 2735891406, 2750483196, 2764518903, 2854316790, 2915768430

Offset: 1

Colin Barker, Apr 29 2014

The first term having a repeated digit is a(83) = 10075823946.

Superset of A115940. - R. J. Mathar, May 02 2014

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000

Cf. A045914, A062691, A162304, A241787, A241788, A241789, A241790, A241791, A241792, A000217.

Select[Table[(n(n+1))/2,{n,45000,100000}],Min[DigitCount[#]]>0&] (* Harvey P. Dale, Jul 26 2014 *)

s=[]; for(n=0, 100000, if(#vecsort(eval(Vec(Str(n*(n+1)/2))), , 8)==10, s=concat(s, n*(n+1)/2))); s

cp's OEIS Frontend

A062691 Triangular numbers that contain exactly 2 different digits.

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A118668 Number of distinct digits needed to write the n-th triangular number in decimal representation.

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A213516 Triangular numbers having only 1 or 2 different digits in base 10.

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A241787 Triangular numbers which have one or more occurrences of exactly four different digits.

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A241788 Triangular numbers which have one or more occurrences of exactly five different digits.

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A241789 Triangular numbers which have one or more occurrences of exactly six different digits.

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A241790 Triangular numbers which have one or more occurrences of exactly seven different digits.

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A241791 Triangular numbers which have one or more occurrences of exactly eight different digits.

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