A043537 -id:A043537 - cp's OEIS frontend

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

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 *)

A076493 Number of common (distinct) decimal digits of n and n^2.

Original entry on oeis.org

1, 1, 0, 0, 0, 1, 1, 0, 0, 0, 2, 1, 1, 1, 1, 1, 1, 0, 0, 1, 1, 1, 0, 1, 0, 2, 1, 2, 1, 0, 1, 1, 1, 0, 0, 1, 1, 1, 0, 0, 1, 1, 1, 1, 0, 1, 1, 0, 1, 1, 2, 1, 1, 0, 0, 1, 1, 0, 0, 0, 2, 1, 0, 2, 2, 1, 1, 0, 1, 1, 1, 1, 0, 1, 2, 1, 2, 0, 1, 0, 1, 1, 1, 1, 0, 1, 1, 1, 0, 1, 1, 1, 0, 1, 0, 2, 2, 1, 1, 1, 2, 2, 2, 2, 2, 3, 2, 1, 1, 1

Offset: 0

Labos Elemer, Oct 21 2002

a(A029783(n)) = 0, a(A189056(n)) > 0; 0 <= a(n) <= 10, see example for first occurrences. [Reinhard Zumkeller, Apr 16 2011]

			a(2) = #{} = 0: 2^2 = 4;
a(0) = #{0} = 1: 0^2 = 0;
a(10) = #{0,1} = 2: 10^2 = 100;
a(105) = #{0,1,5} = 3: 105^2 = 11025;
a(1025) = #{0,1,2,5} = 4: 1025^2 = 1050625;
a(10245) = #{0,1,2,4,5} = 5: 10245^2 = 104960025;
a(102384) = #{0,1,2,3,4,8} = 6: 102384^2 = 10482483456;
a(1023456789) = #{0 .. 9} = 10: 1023456789^2 = 1047463798950190521.

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

Cf. A006880, A076489-A076491.

Cf. A000290, A043537.

import Data.List (intersect, nub)
import Data.Function (on)
a076493 n = length $ (intersect `on` nub . show) n (n^2)
-- Reinhard Zumkeller, Apr 16 2011

Table[Length[Intersection[IntegerDigits[n], IntegerDigits[n^2]]], {n, 1, 100}]

Initial 1 added and offset adjusted by Reinhard Zumkeller, Apr 16 2011

A101594 Numbers with exactly two distinct decimal digits, neither of which is 0.

Original entry on oeis.org

12, 13, 14, 15, 16, 17, 18, 19, 21, 23, 24, 25, 26, 27, 28, 29, 31, 32, 34, 35, 36, 37, 38, 39, 41, 42, 43, 45, 46, 47, 48, 49, 51, 52, 53, 54, 56, 57, 58, 59, 61, 62, 63, 64, 65, 67, 68, 69, 71, 72, 73, 74, 75, 76, 78, 79, 81, 82, 83, 84, 85, 86, 87, 89, 91, 92, 93, 94, 95, 96, 97, 98, 112, 113, 114, 115, 116, 117, 118, 119, 121, 122, 131

Offset: 1

David Wasserman, Dec 07 2004

First differs from A125290 at a(83) = 131 != 123 = A101594(83). - Michael S. Branicky, Dec 13 2021

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
Index entries for 10-automatic sequences.

Cf. A083985, A031955, A004719, A043537, A168046.

Subsequence of A052382, A125290.

a101594 n = a101594_list !! (n-1)
a101594_list = filter ((== 2) . a043537) a052382_list
-- Reinhard Zumkeller, Jun 18 2013

Select[Range[200], FreeQ[#, 0] && Length[Union[#]] == 2 & [IntegerDigits[#]] &] (* Paolo Xausa, May 06 2024 *)

def ok(n): s = set(str(n)); return len(s) == 2 and "0" not in s
print([k for k in range(132) if ok(k)]) # Michael S. Branicky, Dec 13 2021

A168046(a(n)) * A043537(A004719(a(n))) = 2. - Reinhard Zumkeller, Jun 18 2013

A125289 Numbers with unique nonzero digit in decimal representation.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 20, 22, 30, 33, 40, 44, 50, 55, 60, 66, 70, 77, 80, 88, 90, 99, 100, 101, 110, 111, 200, 202, 220, 222, 300, 303, 330, 333, 400, 404, 440, 444, 500, 505, 550, 555, 600, 606, 660, 666, 700, 707, 770, 777, 800, 808, 880, 888, 900, 909

Offset: 1

Reinhard Zumkeller, Nov 26 2006

A043537(a(n)) <= 2.
A043537(A004719(a(n))) = 1: A004719(a(n)) is a repdigit number, see A010785;
also numbers having exactly one partition into digit values of their decimal representations: A061827(a(n))=1.

Rémy Sigrist, Table of n, a(n) for n = 1..9207
Eric Weisstein's World of Mathematics, Repeating Decimal
Eric Weisstein's World of Mathematics, Repdigit

Cf. A125292.

is(n, base=10) = #Set(select(sign, digits(n, base)))==1 \\ Rémy Sigrist, Mar 28 2020

a(n,base=10) = { for (w=0, oo, if (n<=(base-1)*2^w, my (d=1+(n-1)\2^w, k=2^w+(n-1)%(2^w)); return (d*fromdigits(binary(k), base)), n -= (base-1)*2^w)) } \\ Rémy Sigrist, Mar 28 2020

A125289_list = [n for n in range(10**4) if len(set(str(n))-{'0'})==1]
# Chai Wah Wu, Jan 04 2015

from itertools import count, product, islice
def A125289_gen(): # generator of terms
    yield from (int(d+''.join(m)) for l in count(0) for d in '123456789' for m in product('0'+d,repeat=l))
A125289_list = list(islice(A125289_gen(),20)) # Chai Wah Wu, Mar 14 2025

A342441 a(1) = 1; for n > 1, a(n) is the least positive integer not occurring earlier such that a(n-1)+a(n) shares no digit with either a(n-1) or a(n).

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 11, 13, 14, 15, 17, 16, 18, 12, 21, 19, 25, 22, 23, 26, 24, 27, 28, 29, 31, 33, 32, 34, 35, 36, 38, 39, 41, 42, 43, 37, 44, 45, 46, 47, 48, 51, 149, 53, 49, 52, 54, 55, 56, 57, 59, 58, 63, 64, 65, 66, 67, 68, 62, 69, 72, 73, 75, 85, 77, 74, 76, 78, 82, 79

Offset: 1

Scott R. Shannon, Mar 12 2021

No term can end in 0 as that would result in the last digit of a(n-1) being the same as the last digit of a(n-1)+a(n).

			a(2) = 2 as a(1)+2 = 1+2 = 3 which shares no digit with a(1) = 1 or 2.
a(10) = 11 as a(9)+11 = 9+11 = 20 which shares no digit with a(9) = 9 or 11. Note that the first number skipped is 10 as 9+10 = 19 which shares a digit with 9.
a(11) = 13 as a(10)+13 = 11+13 = 24 which shares no digit with a(10) = 11 or 13. Note that the number 12 is skipped as 11+12 = 23 which shares a digit with 12.

Scott R. Shannon, Image of the first 100000 terms. The green line is a(n) = n.

Cf. A342442 (multiplication), A276633, A010784, A043537, A043096, A338466, A336285.

Block[{a = {1}, m = {1}, d, s, k}, Do[k = 2; While[Nand[FreeQ[a, k], ! IntersectingQ[Set[d, IntegerDigits[k]], Set[s, IntegerDigits[a[[-1]] + k]]], ! IntersectingQ[s, m]], k++]; AppendTo[a, k]; Set[m, d], 72]; a] (* Michael De Vlieger, Mar 20 2021 *)

def aupton(terms):
  alst, aset = [1], {1}
  while len(alst) < terms:
    an, anm1_digs = 2, set(str(alst[-1]))
    while True:
      while an in aset: an += 1
      if (set(str(an)) | anm1_digs) & set(str(an+alst[-1])) == set():
        alst.append(an); aset.add(an); break
      an += 1
  return alst
print(aupton(73)) # Michael S. Branicky, Mar 20 2021

A038378 Integers which have more distinct digits than any smaller number.

Original entry on oeis.org

0, 10, 102, 1023, 10234, 102345, 1023456, 10234567, 102345678, 1023456789

Offset: 1

N. J. A. Sloane

Or: Smallest number with exactly n distinct digits. - M. F. Hasler, May 04 2017

S. Colton, Refactorable Numbers - A Machine Invention, J. Integer Sequences, Vol. 2, 1999, #2.
S. Colton, HR - Automatic Theory Formation in Pure Mathematics

Cf. A007809, A043537, A086069.

Prepend[NestList[FromDigits[Append[IntegerDigits[#],Last[Union[IntegerDigits[#]]]+1]]&,10,8],0] (* Ivan N. Ianakiev, May 10 2015 *)
Join[{0},Table[FromDigits[PadRight[{1,0},n,{1,0,2,3,4,5,6,7,8,9}]],{n,2,10}]] (* Harvey P. Dale, Sep 27 2016 *)

A038378(n)=sum(i=1,n--,10^(n-i)*if(i>1,i,10)) \\ M. F. Hasler, May 04 2017

Offset fixed by Reinhard Zumkeller, Aug 25 2009

A137214 a(n) is the number of distinct decimal digits in 2^n.

Original entry on oeis.org

1, 1, 1, 1, 2, 2, 2, 3, 3, 3, 4, 4, 4, 4, 5, 5, 3, 5, 4, 4, 7, 6, 5, 4, 4, 4, 6, 6, 6, 9, 7, 7, 5, 6, 6, 7, 7, 8, 7, 7, 7, 6, 8, 7, 9, 8, 7, 8, 9, 7, 8, 9, 8, 7, 7, 8, 8, 7, 9, 8, 9, 9, 9, 9, 9, 9, 8, 9, 10, 9, 10, 7, 9, 8, 9, 9, 9, 8, 9, 10, 9, 9, 10, 9, 10, 9, 9, 10, 10, 10, 9, 8, 9, 9, 10, 10, 10, 10, 10

Offset: 0

Ctibor O. Zizka, Mar 06 2008

Appears to be all 10's starting at a(169). - T. D. Noe, Apr 01 2014

			a(16) = 3 because 2^16 = 65536, which contains 3 distinct decimal digits [3,5,6].

T. D. Noe, Table of n, a(n) for n = 0..1000

Cf. A043562, A043536.

A043537 := proc(n) nops(convert(convert(n,base,10),set)) ; end: A137214 := proc(n) A043537(2^n) ; end: seq(A137214(n),n=0..120) ; # R. J. Mathar, Mar 16 2008
a:=proc(n) options operator, arrow: nops(convert(convert(2^n,base,10),set)) end proc: seq(a(n),n=0..80); # Emeric Deutsch, Apr 02 2008

Table[Length[Union[IntegerDigits[2^n]]], {n, 0, 100}] (* T. D. Noe, Apr 01 2014 *)

def a(n): return len(set(str(2**n)))
print([a(n) for n in range(99)]) # Michael S. Branicky, Jul 23 2021

a(n) = A043537(2^n). - R. J. Mathar, Mar 16 2008

More terms from R. J. Mathar and Emeric Deutsch, Mar 16 2008

A336285 a(0) = 0; for n > 0, a(n) is the least positive integer not occurring earlier such that the digits in a(n-1)+a(n) are all distinct.

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 7, 6, 8, 9, 10, 11, 12, 13, 14, 15, 16, 18, 17, 19, 20, 21, 22, 23, 24, 25, 26, 27, 29, 28, 30, 31, 32, 33, 34, 35, 36, 37, 38, 40, 39, 41, 42, 43, 44, 45, 46, 47, 48, 49, 53, 50, 52, 51, 54, 55, 65, 58, 62, 61, 59, 64, 56, 67, 57, 63, 60, 66, 68, 69, 70, 72, 71, 74, 73, 75, 77

Offset: 0

Rémy Sigrist, Jul 22 2020

In other words, for any n > 0, a(n) + a(n+1) belongs to A010784.

The sequence is finite since there are only a finite number of positive integers with distinct digits, see A010784, although the exact number of terms is currently unknown.

			The first terms, alongside a(n) + a(n+1), are:
  n   a(n)  a(n)+a(n+1)
  --  ----  -----------
   0     0            1
   1     1            3
   2     2            5
   3     3            7
   4     4            9
   5     5           12
   6     7           13
   7     6           14
   8     8           17
   9     9           19
  10    10           21

Rémy Sigrist, Table of n, a(n) for n = 0..10000
Scott R. Shannon, Image of the first 1000000 terms. The green line is a(n) = n.

Cf. A342383, A338466, A322845, A010784, A043537, A043096, A276633, A002378.

s=0; v=1; for (n=1, 67, print1 (v", "); s+=2^v; for (w=1, oo, if (!bittest(s, w) && #(d=digits(v+w))==#Set(d), v=w; break)))

def agen():
  alst, aset, min_unused = [0], {0}, 1
  yield 0
  while True:
    an = min_unused
    while True:
      while an in aset: an += 1
      t = str(alst[-1] + an)
      if len(t) == len(set(t)):
        alst.append(an); aset.add(an); yield an
        if an == min_unused: min_unused = min(set(range(max(aset)+2))-aset)
        break
      an += 1
g = agen()
print([next(g) for n in range(77)]) # Michael S. Branicky, Mar 11 2021

a(0)=0 added by N. J. A. Sloane, Mar 14 2021

A338466 a(0) = 0; for n > 0, a(n) is the least positive integer not occurring earlier such that the digits in a(n-1)*a(n) are all distinct.

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 12, 11, 13, 14, 15, 16, 19, 18, 17, 20, 21, 22, 23, 26, 24, 27, 25, 29, 28, 30, 31, 33, 32, 39, 34, 37, 35, 36, 38, 40, 41, 43, 42, 45, 44, 47, 50, 49, 52, 48, 55, 46, 51, 53, 56, 54, 57, 60, 58, 62, 59, 66, 61, 64, 63, 65, 71, 70, 67, 69, 68, 72, 74, 73, 77, 79

Offset: 0

Scott R. Shannon, Mar 09 2021

The sequence is finite, the 71782nd term being a(71781) = 50005 beyond which no number exists that has not occurred earlier such that 50005*a(n) has distinct digits. The maximum term is a(71428) = 175446.

			a(1) = 1 as a(0)*1 = 0*1 = 0 which has one distinct digit 0.
a(10) = 10 as a(9)*10 = 9*10 = 90 which has two distinct digits 9 and 0.
a(11) = 12 as a(10)*12 = 10*12 = 120 which has three distinct digits. Note that 11 is the first skipped number as 10*11 = 110 which has 1 as a duplicate digit.
a(12) = 11 as a(11)*11 = 12*11 = 132 which has three distinct digits.

Scott R. Shannon, Image of the 71782 terms. The green line is a(n) = n.

Cf. A342382, A010784, A043537, A043096, A276633, A002378.

Offset corrected by N. J. A. Sloane, Jun 16 2021

A342442 a(1) = 2; for n > 1, a(n) is the least positive integer not occurring earlier such that a(n-1)*a(n) shares no digit with either a(n-1) or a(n).

Original entry on oeis.org

2, 3, 4, 5, 6, 7, 8, 9, 42, 14, 17, 18, 15, 16, 13, 19, 32, 22, 23, 26, 29, 12, 25, 24, 34, 27, 33, 36, 39, 43, 37, 38, 28, 47, 44, 45, 46, 63, 66, 65, 48, 49, 62, 55, 54, 35, 174, 53, 76, 57, 56, 59, 52, 58, 64, 92, 74, 68, 78, 72, 77, 67, 73, 83, 69, 79, 84, 75, 88, 113, 183, 138, 149, 148

Offset: 1

Scott R. Shannon, Mar 12 2021

No term can end in 0 or 1 as that would result in the last digit of a(n-1)*a(n) being the same as a(n)'s last digit. The majority of terms appear to grow linearly with n but occasional large spikes in the values also occur, e.g. a(47888) = 425956849. See the examples. It is unknown if the sequence is infinite.

			a(2) = 3 as a(1)*3 = 2*3 = 6 which shares no digit with a(1) = 2 or 3.
a(9) = 42 as a(8)*42 = 9*42 = 378 which shares no digit with a(8) = 9 or 42.
a(10) = 14 as a(9)*14 = 42*14 = 588 which shares no digit with a(9) = 42 or 14.
a(47888) = 425956849 as a(47887)*425956849 = 258649*425956849 = 110173313037001 which shares no digit with a(47887) = 258649 or 425956849.

Daniel Mondot, Table of n, a(n) for n = 1..10000

Cf. A342441 (addition), A342755, A276633, A010784, A043537, A043096, A338466, A336285.

def aupton(terms):
  alst, aset = [2], {2}
  while len(alst) < terms:
    an, anm1_digs = 2, set(str(alst[-1]))
    while True:
      while an in aset: an += 1
      if (set(str(an)) | anm1_digs) & set(str(an*alst[-1])) == set():
        alst.append(an); aset.add(an); break
      an += 1
  return alst
print(aupton(74)) # Michael S. Branicky, Mar 20 2021

cp's OEIS Frontend

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

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A076493 Number of common (distinct) decimal digits of n and n^2.

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A101594 Numbers with exactly two distinct decimal digits, neither of which is 0.

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Haskell

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Formula

A125289 Numbers with unique nonzero digit in decimal representation.

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PARI

PARI

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Python

A342441 a(1) = 1; for n > 1, a(n) is the least positive integer not occurring earlier such that a(n-1)+a(n) shares no digit with either a(n-1) or a(n).

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A038378 Integers which have more distinct digits than any smaller number.

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PARI

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A137214 a(n) is the number of distinct decimal digits in 2^n.

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Maple