A118668 -id:A118668 - cp's OEIS frontend

A043537 Number of distinct base-10 digits of n.

1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1, 2, 2

Offset: 1

Clark Kimberling

a(A000079(A130694(n))) = 10. - Reinhard Zumkeller, Jul 29 2007
a(A000290(A016070(n))) = 2. - Reinhard Zumkeller, Aug 05 2010
a(n) = 10 for almost all n. - Charles R Greathouse IV, Oct 02 2013

R. Zumkeller, Table of n, a(n) for n = 1..10000

Cf. A047726, A118668, A119797, A119798, A119799, A119823, A038378, A076493, A055642, A178788.

import Data.List (nub)
a043537 = length . nub . show  -- Reinhard Zumkeller, Apr 16 2011

A043537 := proc(n)
        convert(convert(n,base,10),set) ;
        nops(%) ;
end proc: # R. J. Mathar, Dec 22 2012

Count[DigitCount@ #, n_ /; n > 0] & /@ Range@ 120 (* Michael De Vlieger, Aug 29 2015 *)

a(n)=#vecsort(digits(n),,8) \\ Charles R Greathouse IV, Oct 02 2013

def A043537(n): return len(set(str(n))) # Dimiter Skordev, Oct 02 2021

A045914 Triangular numbers with all digits the same.

Original entry on oeis.org

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

Offset: 1

Felice Russo

Escott (1905) proved that there are no more terms with fewer than 30 digits. The complete proof that there are no more terms was given by Ballew and Weger (1972). - Amiram Eldar, Jan 22 2022

Hercher and Fegert pointed out that the proof by Ballew and Weger was flawed, and provided an alternative proof (2025). - Seiichi Azuma, Mar 25 2025

L. E. Dickson, History of the Theory of Numbers, Vol. II, p. 33, Chelsea NY, 1952.
E. B. Escott, Math. Quest. Educational Times, New Series, Vol. 8 (1905), pp. 33-34. - N. J. A. Sloane, Mar 31 2014

David W. Ballew and Ronald C. Weger, Triangular Numbers with Repeated Digits, Proc. S. D. Acad. Sci., Vol. 51 (1972), pp. 52-55.
David W. Ballew and Ronald C. Weger, Repdigit triangular numbers, J. Rec. Math., Vol. 8, No. 2 (1975-76), pp. 96-98.
Christian Hercher and Karl Fegert, Triangular Numbers With a Single Repeated Digit, Journal of Integer Sequences, Vol. 28 (2025), Article 25.2.1.
Bir Kafle, Florian Luca and Alain Togbé, Triangular Repblocks, Fibonacci Quart., Vol. 56, No. 4 (2018), pp. 325-328.
C. E. Youngman, Problem 15648, Educational Times, Vol. 58, 1905, p. 87; with a solution by E. B. Escott.

Cf. A213516 (triangular numbers having only 1 or 2 different digits).

Cf. A118668.

Select[Union[Flatten[Table[FromDigits[PadRight[{},n,k]],{n,3},{k,0,9}]]],OddQ[ Sqrt[8#+1]]&] (* Harvey P. Dale, Feb 11 2020 *)

A118668(a(n)) = 1. - Reinhard Zumkeller, Jul 11 2015

0 inserted by T. D. Noe, Jun 22 2012

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

Original entry on oeis.org

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

A255678 Smallest number m needing exactly n distinct digits in decimal representation of m*(m+1)/2.

Original entry on oeis.org

0, 4, 14, 45, 143, 452, 1431, 4549, 14334, 46097

Offset: 1

Reinhard Zumkeller, Jul 11 2015

A118668(a(n)) = n and A118668(m) <> n for m < a(n).

Cf. A118668, A260214 (the resulting triangular numbers).

a255678 n = head $ filter ((== n) . a118668) [0..]

A115939 Numbers k such that the k-th triangular number contains each of the 10 decimal digits exactly once.

Original entry on oeis.org

46097, 49796, 50139, 55151, 55484, 56520, 58050, 62495, 62567, 62900, 66294, 68805, 69542, 70766, 72594, 73737, 73971, 74168, 74357, 75555, 76364, 77805, 78686, 78848, 84555, 85959, 86076, 87263, 87669, 88406, 89883, 90287, 90297

Offset: 1

Giovanni Resta, Feb 06 2006

There are 82 such numbers, the largest being 138959.

A118668(a(n)) = 10. - Reinhard Zumkeller, Jul 11 2015

			T(75555) = 2854316790.

Zak Seidov, Table of n, a(n) for n = 1..82 (full sequence)

Cf. A000217, A115940, A118668.

Select[Range[141000], Sort[IntegerDigits[# (# + 1)/2]] == Range[0, 9] &] (* Giovanni Resta, Mar 19 2013 *)
(Sqrt[8#+1]-1)/2&/@Select[FromDigits/@Select[Permutations[Range[0,9]], #[[1]]>0&],OddQ[Sqrt[8#+1]]&] (* Harvey P. Dale, Sep 19 2016 *)

A260214 a(n) is the first triangular number whose decimal representation uses n distinct digits.

Original entry on oeis.org

0, 10, 105, 1035, 10296, 102378, 1024596, 10348975, 102738945, 1062489753

Offset: 1

Anders Hellström, Jul 19 2015

Anders Hellström, Table of n, a(n) for n = 1..10

Cf. A000217, A043537, A118668, A255678.

# (# + 1)/2 & /@ {0, 4, 14, 45, 143, 452, 1431, 4549, 14334, 46097} (* terms in A255678 *) (* Robert G. Wilson v, Jul 21 2015 *)

number_of_distinct_digits(m) = if(m==0,1,#vecsort(digits(m), , 8))
a(n)=my(m=0); while(!(number_of_distinct_digits(m*(m+1)/2)==n), m++); m*(m+1)/2;
first(m)=vector(m, n, a(n)) \\m<=10 /* Anders Hellström, Aug 03 2015 */

a(n) = A255678(n)*(A255678(n)+1)/2.

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

A043537 Number of distinct base-10 digits of n.

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A045914 Triangular numbers with all digits the same.

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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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A255678 Smallest number m needing exactly n distinct digits in decimal representation of m*(m+1)/2.

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Haskell

A115939 Numbers k such that the k-th triangular number contains each of the 10 decimal digits exactly once.

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Mathematica

A260214 a(n) is the first triangular number whose decimal representation uses n distinct digits.

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Mathematica

PARI

Formula

A352057 Triangular numbers whose nonzero digits are all the same.

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