A062691 -id:A062691 - cp's OEIS frontend

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

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

A325907 a(n) = ( (-1)^n * Sum_{k=0..n-2} (-1)^k*10^(2^k) + 10^(2^(n-1)) - ((-1)^n+3)/2 )/3.

Original entry on oeis.org

3, 36, 3363, 33336636, 3333333366663363, 33333333333333336666666633336636, 3333333333333333333333333333333366666666666666663333333366663363

Offset: 1

Seiichi Manyama, Sep 08 2019

nonn

All terms are elements of A213517.

			              36 =        -3 - 1 +        4 * 10^1.
            3363 =       -36 - 1 +       34 * 10^2.
        33336636 =     -3363 - 1 +     3334 * 10^4.
3333333366663363 = -33336636 - 1 + 33333334 * 10^8.
------------------------------------------------------
T(n) = n*(n+1)/2.
               T(3) =                               6.
              T(36) =                             666.
            T(3363) =                         5656566.
        T(33336636) =                 555665666566566.
T(3333333366663363) = 5555555666655656666556566566566.

Seiichi Manyama, Table of n, a(n) for n = 1..10

Cf. A000217, A062691, A093137, A119219, A213517, A309597, A325906.

a[n_] := ((-1)^n * Sum[(-1)^k * 10^(2^k), {k, 0, n - 2}] + 10^(2^(n - 1)) - ((-1)^n + 3)/2)/3; Array[a, 7] (* Amiram Eldar, May 07 2021 *)

{a(n) = ((-1)^n*sum(k=0, n-2, (-1)^k*10^2^k)+10^2^(n-1)-((-1)^n+3)/2)/3}

a(n) = 3 * A325906(n).

a(n) = -a(n-1) - 1 + A093137(2^(n-2)) * 10^(2^(n-2)).

A213518 Numbers k such that the triangular number k*(k+1)/2 has 2 different digits in base 10.

Original entry on oeis.org

4, 5, 6, 7, 8, 9, 12, 13, 18, 24, 34, 44, 58, 66, 77, 100, 101, 105, 109, 114, 132, 141, 363, 666, 714, 816, 1000, 1095, 1287, 1332, 1541, 3363, 6666, 10000, 10114, 13332, 66666, 100000, 133332, 666666, 1000000, 1333332, 6666666, 10000000, 13333332, 33336636, 66666666, 100000000

Offset: 1

T. D. Noe, Jun 22 2012

The list of triangular numbers containing only one digit (A045914) is finite. This list is infinite because numbers like 133332, 666666, and 1000000 occur an infinite number of times.
A118668(a(n)) = 2. - Reinhard Zumkeller, Jul 11 2015
For n > 2, A325907(n) is a term. - Seiichi Manyama, Sep 15 2019

David Radcliffe, Table of n, a(n) for n = 1..116 (terms 1..51 from Seiichi Manyama, terms 52..60 from T. D. Noe).

Cf. A062691 (the corresponding triangular numbers), A213516, A213517, A325907.
Cf. A118668.
Cf. A187127.

a213518 n = a213518_list !! (n-1)
a213518_list = filter ((== 2) . a118668) [0..]
-- Reinhard Zumkeller, Jul 11 2015

t = {}; Do[tri = n*(n+1)/2; If[Length[Union[IntegerDigits[tri]]] == 2, AppendTo[t, n]], {n, 10^5}]; t

for(k=0, 1e8, if(#Set(digits(k*(k+1)/2))==2, print1(k", "))) \\ Seiichi Manyama, Sep 15 2019

a(45)-a(48) from Seiichi Manyama, Sep 15 2019

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

cp's OEIS Frontend

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

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

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A325907 a(n) = ( (-1)^n * Sum_{k=0..n-2} (-1)^k*10^(2^k) + 10^(2^(n-1)) - ((-1)^n+3)/2 )/3.

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