A031955 -id:A031955 - cp's OEIS frontend

A031953 Numbers with exactly two distinct base-8 digits.

8, 10, 11, 12, 13, 14, 15, 16, 17, 19, 20, 21, 22, 23, 24, 25, 26, 28, 29, 30, 31, 32, 33, 34, 35, 37, 38, 39, 40, 41, 42, 43, 44, 46, 47, 48, 49, 50, 51, 52, 53, 55, 56, 57, 58, 59, 60, 61, 62, 64, 65, 72, 74, 75, 76, 77, 78, 79, 81, 82, 89, 91, 97, 100, 105

Offset: 1

Clark Kimberling

Index entries for 2-automatic sequences.

Cf. A031948 - A031955.

Select[Range[100],Length[Union[IntegerDigits[#,8]]]==2&] (* Harvey P. Dale, Feb 15 2019 *)

is(n)=#Set(digits(n, 8))==2 \\ Charles R Greathouse IV, Feb 15 2017

def ok(n): return len(set(oct(n)[2:])) == 2
print(list(filter(ok, range(106)))) # Michael S. Branicky, Aug 10 2021

Name edited by Michael S. Branicky, Aug 10 2021

A380974 Numbers k such that k*(k-1) is composed of exactly two different decimal digits.

Original entry on oeis.org

4, 5, 6, 7, 8, 9, 10, 11, 17, 24, 25, 32, 34, 67, 75, 78, 100, 101, 142, 167, 334, 667, 1000, 1001, 1667, 3334, 6667, 10000, 10001, 16667, 33334, 66667, 100000, 100001, 166667, 333334, 666667, 1000000, 1000001, 1666667, 3333334, 6666667, 10000000, 10000001, 16666667, 33333334, 66666667, 100000000

Offset: 1

Robert Israel, Feb 11 2025

Numbers k such that A002378(k-1) is in A031955.
Conjecture: all terms >= 334 are of the form 10...0, 10...01, 16...67, 3...34, or 6...67.
Last decimal digit of a(n)*(a(n)-1) is either 0, 2 or 6. - Chai Wah Wu, Feb 19 2025

			a(10) = 23 is a term because 23 * 24 = 552 contains two different digits 2 and 5.

Michael S. Branicky, Table of n, a(n) for n = 1..117

Cf. A002378, A031955, A016069, A380984.

select(k -> nops(convert(convert(k*(k+1),base,10),set)) = 2, [$1..10^6]);

Select[Range[10^7],Length[Union[IntegerDigits[#*(#-1)]]]==2&] (* James C. McMahon, Feb 13 2025 *)

isok(k) = #Set(digits(k*(k-1))) == 2; \\ Michel Marcus, Feb 11 2025

from math import isqrt
from itertools import count, combinations, product, islice
def A380974_gen(): # generator of terms
    for n in count(1):
        c = []
        for a in combinations('0123456789',2):
            if '0' in a or '2' in a or '6' in a:
                for b in product(a,repeat=n):
                    if b[0] != '0' and b[-1] in {'0','2','6'} and b != (a[0],)*n and b != (a[1],)*n:
                        m = int(''.join(b))
                        q = isqrt(m)
                        if q*(q+1)==m:
                            c.append(q+1)
        yield from sorted(c)
A380974_list = list(islice(A380974_gen(),30)) # Chai Wah Wu, Feb 19 2025

Conjectured: for k >= 0,
  a(20 + 5*k) = (10^(3+k) + 2)/6,
  a(21 + 5*k) = (10^(3+k) + 2)/3,
  a(22 + 5*k) = (2*10^(3+k)+1)/3,
  a(23 + 5*k) = 10^(3+k),
  a(24 + 5*k) = 10^(3+k) + 1.
Conjectured G.f.: (4*x + 5*x^2 + 6*x^3 + 7*x^4 + 8*x^5 - 35*x^6 - 45*x^7 - 55*x^8 - 60*x^9 - 64*x^10 - 34*x^11 - 28*x^12 - 27*x^13 - 50*x^14 - 109*x^15 - 107*x^16 - 152*x^17 - 163*x^18 - 425*x^19 - 418*x^20 - 274*x^21 - 113*x^22 + 229*x^23 + 109*x^24 + 580*x^25 + 440*x^26 + 330*x^27 + 10*x^28 + 410*x^29)/(1 - 11 * x^5 + 10 * x^10).

A334963 a(n) is the least positive multiple of n that has at most two distinct digits.

Original entry on oeis.org

1122, 515, 1144, 525, 212, 535, 1188, 545, 110, 111, 112, 113, 114, 115, 116, 117, 118, 119, 600, 121, 122, 3444, 744, 500, 252, 889, 4224, 774, 10010, 131, 660, 133, 4422, 4455, 272, 411, 414, 556, 700, 141, 994, 858, 144, 3335, 292, 441, 444, 447, 300, 151, 2888

Offset: 102

David A. Corneth, May 17 2020

			a(102) = 1122 as 1122 = 11*102 is the least multiple of 102 that has at most 2 distinct digits.

David A. Corneth, Table of n, a(n) for n = 102..10101

Cf. A004290, A031955, A181060.

a[n_] := Module[{k = n}, While[Length @ Select[DigitCount[k, 10], # > 0 &] > 2, k += n]; k]; Array[a, 51, 102] (* Amiram Eldar, May 21 2020 *)

a(n) = for(i = 1, oo, if(#Set(digits(i*n))<3, return(i*n)))

a(n) <= A004290(n).

a(n) = n if n is in A031955. - Bernard Schott, May 17 2020

A380997 a(n) is the least number with exactly 2 different decimal digits that is a multiple of n.

Original entry on oeis.org

10, 10, 12, 12, 10, 12, 14, 16, 18, 10, 110, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 110, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 330, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 220, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 110, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 330, 67, 68, 69, 70, 71, 72, 73

Offset: 1

Robert Israel, Feb 11 2025

a(n) is the least multiple of n that is in A031955.

			a(22) = 110 because 110 has two different decimal digits 0 and 1, is a multiple of 22, and no smaller multiple of 22 works.

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A031955.

f:= proc(n) local d,i,s,S;
       S:= select(t -> nops(convert(convert(t*n mod 100,base,10),set)) <= 2, [$1..99] );
       for d from 3 do
         S:= select(s -> nops(convert(convert(s*n mod 10^d,base,10),set)) <= 2,
               [seq(seq(s+i*10^(d-1),s = S),i=0..9)]);
         for s in S do if nops(convert(convert(s*n,base,10),set)) = 2  then return s*n fi od;
       od;
end proc:
map(f, [$1..1000]);

a[n_]:=Module[{k=n}, While[Length[DeleteDuplicates[IntegerDigits[k]]]!=2, k+=n]; k]; Array[a,73] (* Stefano Spezia, Feb 13 2025 *)

a(n) = my(k=n); while(#Set(digits(k)) != 2, k+=n); k; \\ Michel Marcus, Feb 13 2025

A288040 Integers whose number of distinct decimal digits is prime.

Original entry on oeis.org

10, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76, 78, 79, 80, 81, 82, 83, 84, 85, 86, 87, 89, 90, 91, 92, 93, 94, 95, 96, 97, 98, 100, 101

Offset: 1

Jonathan Frech, Jun 04 2017

Differs from A139819 (which contains, for example, 1234, a number with 4 distinct decimal digits). - R. J. Mathar, Jun 14 2017

Union of A031955 and A031962 and ....

Select[Range@ 101, PrimeQ@ Count[DigitCount[#], ?(# != 0 &)] &] (* _Michael De Vlieger, Jun 06 2017 *)

isok(m) = isprime(#Set(digits(m))); \\ Michel Marcus, May 10 2020

from sympy import isprime
print([n for n in range(1, 100) if isprime(len(set(str(n))))])

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A031953 Numbers with exactly two distinct base-8 digits.

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A380974 Numbers k such that k*(k-1) is composed of exactly two different decimal digits.

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A334963 a(n) is the least positive multiple of n that has at most two distinct digits.

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A380997 a(n) is the least number with exactly 2 different decimal digits that is a multiple of n.

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Maple

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A288040 Integers whose number of distinct decimal digits is prime.

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Mathematica

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Python