A235154 -id:A235154 - cp's OEIS frontend

A031955 Numbers with exactly two distinct base-10 digits.

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, 110, 112, 113, 114, 115, 116, 117, 118, 119, 121, 122, 131, 133, 141, 144, 151, 155, 161, 166

Offset: 1

Clark Kimberling

The three-digit terms are given by A210666(1,...,244). For numbers with exactly two distinct (but unspecified) digits in other bases, see A031948-A031954. For numbers made of two *given* digits, see A007088 (digits 0 & 1), A007931 (digits 1 & 2), A032810 (digits 2 & 3), A032834 (digits 3 & 4), A256290 (digits 4 & 5), A256291 (digits 5 & 6), A256292 (digits 6 & 7), A256340 (digits 7 & 8), A256341 (digits 8 & 9), and A032804-A032816 (in other bases). - M. F. Hasler, Apr 04 2015
A235154 is a subsequence. - Altug Alkan, Dec 03 2015
A235717 is a subsequence. - Robert Israel, Dec 03 2015

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

Different from A029742.

Cf. A043638, A101594, A031948, A031949, A031950, A031951, A031952, A031953, A031954, A235154, A235717.

a031955 n = a031955_list !! (n-1)
a031955_list = filter ((== 2) . a043537) [0..]
-- Reinhard Zumkeller, Feb 05 2012

M:= 5: # to get all terms < 10^M
sort([seq(seq(seq(seq(add(10^(m-j)*`if`(member(j,S2),d2,d1),j=1..m)  ,
  S2 = combinat:-powerset({$2..m}) minus {{}}),
  d2 = {$0..9} minus {d1}), d1 = 1..9), m=2..M)]); # Robert Israel, Dec 03 2015

Select[Range@ 166, Length@ Union@ IntegerDigits@ # == 2 &] (* Michael De Vlieger, Dec 03 2015 *)

is_A031955(n)=#Set(digits(n))==2 \\ M. F. Hasler, Apr 04 2015

def ok(n): return len(set(str(n))) == 2
print(list(filter(ok, range(167)))) # Michael S. Branicky, Oct 12 2021

A043537(a(n)) = 2. - Reinhard Zumkeller, Dec 03 2009

Name edited by Charles R Greathouse IV, Feb 13 2017

A235161 Primes which have one or more occurrences of exactly nine different digits.

Original entry on oeis.org

102345689, 102345697, 102345869, 102346789, 102346879, 102346897, 102346957, 102347689, 102348679, 102348769, 102349867, 102354689, 102354697, 102356489, 102356789, 102356987, 102358769, 102358967, 102364859, 102364879, 102365897, 102365947, 102368459

Offset: 1

Colin Barker, Jan 04 2014

The first term having a repeated digit is 1002346589.

Colin Barker, Table of n, a(n) for n = 1..2000

Cf. A073643, A235154-A235160.

s=[]; forprime(n=100000000, 102400000, if(#vecsort(eval(Vec(Str(n))),,8)==9, s=concat(s, n))); s

A030291 Primes with at most two different digits.

Original entry on oeis.org

2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 101, 113, 131, 151, 181, 191, 199, 211, 223, 227, 229, 233, 277, 311, 313, 331, 337, 353, 373, 383, 433, 443, 449, 499, 557, 577, 599, 661, 677, 727, 733

Offset: 1

Patrick De Geest

The one-digit primes (2, 3, 5, 7) followed by the union of A004022 and A235154. - Jeppe Stig Nielsen, Feb 17 2021

Jeppe Stig Nielsen, Table of n, a(n) for n = 1..21936 (first 1000 terms from T. D. Noe)

Cf. A004022, A032758, A116692, A235154.

Select[Range[1000], PrimeQ[#] && Length[Union[RealDigits[#][[1]]]] <= 2 &]
Select[Prime[Range[200]],Count[DigitCount[#],0]>7&] (* Harvey P. Dale, Jul 14 2017 *)

Offset corrected by Arkadiusz Wesolowski, Sep 13 2011

A235690 Semiprimes which have one or more occurrences of exactly two different digits.

Original entry on oeis.org

10, 14, 15, 21, 25, 26, 34, 35, 38, 39, 46, 49, 51, 57, 58, 62, 65, 69, 74, 82, 85, 86, 87, 91, 93, 94, 95, 115, 118, 119, 121, 122, 133, 141, 155, 161, 166, 177, 202, 221, 226, 262, 299, 303, 323, 334, 335, 339, 355, 377, 393, 411, 422, 445, 446, 447, 454

Offset: 1

Colin Barker, Jan 14 2014

The first term having a repeated digit is 115.

			1000000000010101 is a term because it is made of the digits 0 and 1 and it is the product of the two primes 18463559 and 54160739.

Cf. A235691-A235696.

Cf. A235154-A235161.

Select[Range[454], Length@Union@ IntegerDigits[#] == 2 && Total[Last /@ FactorInteger[#]] == 2 &] (* Giovanni Resta, Jan 14 2014 *)

list(lim)=my(v=List(), t); forprime(p=2, sqrt(lim), t=p; forprime(q=p, lim\t, listput(v, t*q))); vecsort(Vec(v)) \\ From A001358
b=list(10000); s=[]; for(n=1, #b, if(#vecsort(eval(Vec(Str(b[n]))),,8)==2, s=concat(s, b[n]))); s

A157711 Primes made up of 0's and four 1's only.

Original entry on oeis.org

10111, 1011001, 1100101, 10010101, 10100011, 101001001, 1000001011, 1000010101, 1010000011, 1100010001, 10000001101, 10001000011, 10001001001, 10001100001, 10100000011, 10100001001, 11000000101, 11001000001

Offset: 1

Lekraj Beedassy, Mar 04 2009

Intersection of A062339 and A020449. Subsequence of A235154. - Felix Fröhlich, Nov 19 2014

Primes that are the sum of four distinct powers of ten (A038446). - Jeppe Stig Nielsen, May 18 2023

Jeppe Stig Nielsen, Table of n, a(n) for n = 1..10000 (terms 1..1000 from T. D. Noe)
Hans Havermann, The one-millionth term.
Hans Havermann, The two-millionth term.

Cf. A020449, A038446, A062339, A235154, A383675 (number of n-digit terms).

for d from 4 to 18 do for c from 0 to 2^d-1 do bdgs := convert(c,base,2) ; if add(i,i=bdgs) = 3 then p := 10^d+add(op(i,bdgs)*10^(i-1),i=1..nops(bdgs)) ; if isprime(p) then printf("%d,",p) ; fi; fi; od: od: # R. J. Mathar, Mar 06 2009

Flatten[Select[FromDigits/@Permutations[Join[{1,1,1,1},PadRight[{},7,0]]],PrimeQ]] // Union (* Harvey P. Dale, May 09 2019 *)

for(n=0, 10, forprime(p=10^n, (10^(n+1)-1)/9, if(vecmax(digits(p))==1, if(sumdigits(p)==4, print1(p, ", "))))) \\ Felix Fröhlich, Nov 19 2014

my(M=20);for(i=3, M, for(j=2,i-1, for(k=1, j-1, my(p=10^i+10^j+10^k+1); isprime(p)&&print1(p,", ")))) \\ Jeppe Stig Nielsen, May 18 2023

Extended by numerous authors, Mar 06 2009

A235696 Semiprimes which have one or more occurrences of exactly eight different digits.

Original entry on oeis.org

10234569, 10234657, 10234685, 10234687, 10234769, 10234795, 10234859, 10234865, 10234879, 10234957, 10234967, 10235469, 10235479, 10235489, 10235497, 10235679, 10235689, 10235769, 10235789, 10235798, 10235846, 10235847, 10235879, 10235894, 10235947, 10235986

Offset: 1

Colin Barker, Jan 14 2014

Harvey P. Dale, Table of n, a(n) for n = 1..1000

Cf. A235690-A235695.

Cf. A235154-A235161.

Select[Range[10234000,10236000],PrimeOmega[#]==Count[DigitCount[#],0]==2&] (* Harvey P. Dale, Sep 01 2014 *)

list(lim)=my(v=List(), t); forprime(p=2, sqrt(lim), t=p; forprime(q=p, lim\t, listput(v, t*q))); vecsort(Vec(v)) \\ From A001358
b=list(1030000); s=[]; for(n=1, #b, if(#vecsort(eval(Vec(Str(b[n]))),,8)==8, s=concat(s, b[n]))); s

A034845 Primes of the form iii...ijjj...j, i != j.

Original entry on oeis.org

13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 113, 199, 211, 223, 227, 229, 233, 277, 311, 331, 337, 433, 443, 449, 499, 557, 577, 599, 661, 677, 733, 773, 811, 877, 881, 883, 887, 911, 977, 991, 997, 1117, 1777, 1999, 2111

Offset: 1

Erich Friedman

Alois P. Heinz, Table of n, a(n) for n = 1..10034 (first 1000 terms from Sean A. Irvine)

Cf. A235154.

Select[Union[Flatten[Table[FromDigits[Join[PadRight[{},n,m],PadRight[{},k,q]]],{n,3},{m,9},{k,3},{q,{1,3,7,9}}]]],IntegerDigits[#][[1]]!=IntegerDigits[#][[-1]]&&PrimeQ[#]&] (* Harvey P. Dale, Apr 06 2022 *)

A235155 Primes which have one or more occurrences of exactly three different digits.

Original entry on oeis.org

103, 107, 109, 127, 137, 139, 149, 157, 163, 167, 173, 179, 193, 197, 239, 241, 251, 257, 263, 269, 271, 281, 283, 293, 307, 317, 347, 349, 359, 367, 379, 389, 397, 401, 409, 419, 421, 431, 439, 457, 461, 463, 467, 479, 487, 491, 503, 509, 521, 523, 541, 547

Offset: 1

Colin Barker, Jan 04 2014

The first term having a repeated digit is 1009.

Christopher M. Conrey, Table of n, a(n) for n = 1..10000 (first 2000 terms from Colin Barker)
Christopher M. Conrey, MATLAB Program

Cf. A235154, A235156, A235157, A235158, A235159, A235160, A235161.

Cf. A074675.

MATLAB
```
%See Conrey Link
```

Select[Prime[Range[200]],Count[DigitCount[#],0]==7&] (* Harvey P. Dale, Jul 27 2020 *)

s=[]; forprime(n=100, 1000, if(#vecsort(eval(Vec(Str(n))),,8)==3, s=concat(s, n))); s

A235157 Primes which have one or more occurrences of exactly five different digits.

Original entry on oeis.org

10243, 10247, 10253, 10259, 10267, 10273, 10289, 10357, 10369, 10427, 10429, 10453, 10457, 10459, 10463, 10487, 10529, 10567, 10589, 10597, 10627, 10639, 10657, 10687, 10723, 10729, 10739, 10753, 10789, 10837, 10847, 10853, 10859, 10867, 10937, 10957

Offset: 1

Colin Barker, Jan 04 2014

The first term having a repeated digit is 100237.

There are 2,529 5-digit primes in the sequence. Harvey P. Dale, Feb 06 2015

Colin Barker, Table of n, a(n) for n = 1..2500

Cf. A235154-A235156, A235158-A235161.

Cf. A074671.

Select[Prime[Range[1255,1355]],Max[DigitCount[#]]==1&] (* The program is only accurate for 5-digit primes, of which there are 2529 satisfying the definition *) (* Harvey P. Dale, Feb 06 2015 *)

s=[]; forprime(n=10000, 13000, if(#vecsort(eval(Vec(Str(n))),,8)==5, s=concat(s, n))); s

A335843 a(n) is the number of n-digit positive integers with exactly two distinct base 10 digits.

Original entry on oeis.org

0, 81, 243, 567, 1215, 2511, 5103, 10287, 20655, 41391, 82863, 165807, 331695, 663471, 1327023, 2654127, 5308335, 10616751, 21233583, 42467247, 84934575, 169869231, 339738543, 679477167, 1358954415, 2717908911, 5435817903, 10871635887, 21743271855, 43486543791

Offset: 1

Stefano Spezia, Jul 18 2020

a(n) is the number of n-digit numbers in A031955.

			a(1) = 0 since the positive integers must have at least two digits;
a(2) = 81 since #[99] - #[9] - #(11*[9]) = 99 - 9 - 9 = 81;
a(3) = 243 since #[999] - #[99] - #(111*[9]) - #{xyz in N | x,y,z are three different digits with x != 0} = 999 - 99 - 9 - 9*9*8 = 243;
...

Stefano Spezia, Table of n, a(n) for n = 1..3300
Puzzle Critic, Twitter post about the case n = 5, Jul 16 2020.
Index entries for linear recurrences with constant coefficients, signature (3,-2).
Index entries for sequences related to digits.

Cf. A000225, A031955, A337127, A337313.

Cf. A055642, A235154, A235690, A235717.

Mathematica
```
LinearRecurrence[{3,-2},{0,81},31]
```

concat([0],Vec(81*x^2/(1-3*x+2*x^2)+O(x^31)))

O.g.f.: 81*x^2/(1 - 3*x + 2*x^2).
E.g.f.: 81*(exp(x) - 1)^2/2.
a(n) = 3*a(n-1) - 2*a(n-2) for n > 2.
a(n) = 81*(2^(n-1) - 1).
a(n) = 81*A000225(n-1).

a(0) removed by Stefano Spezia, Sep 23 2020

cp's OEIS Frontend

A031955 Numbers with exactly two distinct base-10 digits.

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A235161 Primes which have one or more occurrences of exactly nine different digits.

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A030291 Primes with at most two different digits.

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A235690 Semiprimes which have one or more occurrences of exactly two different digits.

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A157711 Primes made up of 0's and four 1's only.

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A235696 Semiprimes which have one or more occurrences of exactly eight different digits.

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A034845 Primes of the form iii...ijjj...j, i != j.

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A235155 Primes which have one or more occurrences of exactly three different digits.

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