A045914 -id:A045914 - cp's OEIS frontend

A213517 Numbers n such that the triangular number n*(n+1)/2 has only 1 or 2 different digits in base 10.

0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 18, 24, 34, 36, 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

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 133332, 666666, and 1000000 occur an infinite number of times.
A118668(a(n)) <= 2. - Reinhard Zumkeller, Jul 11 2015
A325907(n) is a term. - Seiichi Manyama, Sep 14 2019

Seiichi Manyama, Table of n, a(n) for n = 1..58

Cf. A000217, A045914, A213516, A322570, A325907.

Cf. A118668.

a213517 n = a213517_list !! (n-1)
a213517_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, 0, 10^5}]; t

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

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

A274084 Triangular numbers that are repdigits with length > 2 in some base.

Original entry on oeis.org

15, 21, 91, 171, 666, 703, 820, 1830, 1953, 3003, 3081, 4095, 7140, 7381, 10440, 12720, 14706, 16471, 16653, 18915, 23871, 24976, 30628, 47586, 47895, 48828, 66430, 71631, 79401, 95703, 101475, 104653, 119805, 128778, 148240, 148785, 173166, 191271, 221445

Offset: 1

Robert Israel, Jun 09 2016

Intersection of A000217 and A167782.

Sequence is infinite, e.g. for any k>=2 and j>=1 it contains n*(n+1)/2 where n = ((8j+1)^k-1)/2: this has 2k digits of j in base 8j+1.

			15 = 5*6/2 = 1111_2.
21 = 6*7/2 = 111_4.
91 = 13*14/2 = 111_9.
171 = 18*19/2 = 333_7.

Robert Israel, Table of n, a(n) for n = 1..694

Cf. A000217, A045914, A167782.

N:= 10^9: # to get all entries <= N
S:= {}:
for b from 2 to floor(sqrt(N)) do
  for k from 3 do
     r:= (b^k-1)/(b-1);
     if r > N then break fi;
     for a from 1 to min(b-1, N/r) do
        if issqr(1+8*r*a) then
          S:= S union {r*a}
        fi
     od
  od
od:
sort(convert(S,list));

A343811 Triangular numbers k such that every permutation of the digits of k is a triangular number.

Original entry on oeis.org

0, 1, 3, 6, 10, 55, 66, 666

Offset: 1

Ctibor O. Zizka, Apr 30 2021

I assume this sequence is finite.
From Jon E. Schoenfield, Mar 05 2022: (Start)
If k is a term of this sequence and is not a repdigit, then at least one of its permutations (a triangular number, call it k1) will have a digit d0 in its ones place and a digit d1 != d0 in its tens place, and the number that results from reversing those last two digits will be a triangular number k2 = k1 + 9*(d0 - d1), so the two distinct triangular numbers k2 and k1 will differ by no more than 9*(9-0) = 81. But the j-th triangular number T(j) differs from the nearest other triangular number by T(j) - T(j-1) = j, so if two distinct triangular numbers T(k1) > T(k2) differ by no more than 81, then k1 <= 81. No triangular numbers <= T(81) = 3321 are terms that exceed 666, so if there exists any term > 666, it is a repdigit triangular number.
However, Ballew and Weger proved (see A045914) that the only repdigit triangular numbers are 0, 1, 3, 6, 55, 66, and 666. Thus, 666 is the last term of this sequence. (End)

			10 = 4*5/2, 01 = 1*2/2.

Cf. A000217, A003459, A045914, A077671.

triQ[n_] := IntegerQ @ Sqrt[8*n + 1]; Select[Range[0, 1000], AllTrue[Permutations[ IntegerDigits[#] ], triQ[FromDigits[#1]] &] &] (* Amiram Eldar, Apr 30 2021 *)
pdtQ[n_]:=AllTrue[FromDigits/@Permutations[IntegerDigits[n]],OddQ[ Sqrt[ 8#+1]]&]; Select[Accumulate[Range[0,5000]],pdtQ] (* Requires Mathematica version 10 or later *) (* Harvey P. Dale, Jul 04 2021 *)

A069172 Smallest triangular number such that arithmetic mean of its digits = n (sum of the digits = n times the number of digits).

Original entry on oeis.org

1, 105, 3, 435, 28, 6, 5995, 589598998999878

Offset: 1

Robert G. Wilson v, Apr 09 2002

Because A045914 is complete, this sequence is complete. - Robert G. Wilson v, Oct 30 2013

Cf. A069790.

Do[k = 1; While[ Apply[ Plus, IntegerDigits[k(k + 1)/2]] != n*Floor[ Log[10, k(k + 1)/2] + 1], k++ ]; Print[k(k + 1)/2], {n, 1, 7}]

a(8) from Donovan Johnson, Mar 30 2010

A322045 Triangular numbers that are 2-digit-repblocks.

Original entry on oeis.org

10, 15, 21, 28, 36, 45, 55, 66, 78, 91, 5050, 5151

Offset: 1

N. J. A. Sloane, Dec 07 2018

Bir Kafle, Florian Luca, and Alain Togbé, Triangular repblocks, The Fibonacci Quarterly, 56:4 (2018), 325-328.

Cf. A000217, A010785, A045914.

Select[Union[Flatten[Table[FromDigits[PadRight[{}, n, k], 100], {n, 3}, {k, 10, 99}]]], IntegerQ[Sqrt[8*# + 1]] &] (* Amiram Eldar, Mar 07 2025, after Harvey P. Dale at A045914 *)

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A213517 Numbers n such that the triangular number n*(n+1)/2 has only 1 or 2 different digits in base 10.

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

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A274084 Triangular numbers that are repdigits with length > 2 in some base.

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Maple

A343811 Triangular numbers k such that every permutation of the digits of k is a triangular number.

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Mathematica

A069172 Smallest triangular number such that arithmetic mean of its digits = n (sum of the digits = n times the number of digits).

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Mathematica

Extensions

A322045 Triangular numbers that are 2-digit-repblocks.

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Mathematica