A020450 -id:A020450 - cp's OEIS frontend

A385776 Primes having only {1, 2, 9} as digits.

2, 11, 19, 29, 191, 199, 211, 229, 911, 919, 929, 991, 1129, 1229, 1291, 1999, 2111, 2129, 2221, 2999, 9199, 9221, 9929, 11119, 11299, 12119, 12211, 12911, 12919, 19121, 19211, 19219, 19919, 19991, 21121, 21191, 21211, 21221, 21911, 21929, 21991

Offset: 1

Jason Bard, Jul 09 2025

Supersequence of A020450, A020457, A020460.

Cf. A000040.

[p: p in PrimesUpTo(10^6) | Set(Intseq(p)) subset [1, 2, 9]];

Flatten[Table[Select[FromDigits /@ Tuples[{1, 2, 9}, n], PrimeQ], {n, 7}]]

primes_with(n=50, show=0, L=[1, 2, 9])={for(d=1, 1e9, my(t, u=vector(d, i, 10^(d-i))~); forvec(v=vector(d, i, [1+!(L[1]||(i>1&&i

from gmpy2 import is_prime
from itertools import count, islice, product
def primes_with(digits):  # generator of primes having only set(digits) as digits
    S, E = "".join(sorted(set(digits) - {'0'})), "".join(sorted(set(digits) & set("1379")))
    yield from (p for p in [2, 3, 5, 7] if str(p) in digits)
    yield from (t for d in count(2) for s in S for m in product(digits, repeat=d-2) for e in E if is_prime(t:=int(s+"".join(m)+e)))
print(list(islice(primes_with("129"), 41))) # Michael S. Branicky, Jul 11 2025

A007931 Numbers that contain only 1's and 2's. Nonempty binary strings of length n in lexicographic order.

Original entry on oeis.org

1, 2, 11, 12, 21, 22, 111, 112, 121, 122, 211, 212, 221, 222, 1111, 1112, 1121, 1122, 1211, 1212, 1221, 1222, 2111, 2112, 2121, 2122, 2211, 2212, 2221, 2222, 11111, 11112, 11121, 11122, 11211, 11212, 11221, 11222, 12111, 12112, 12121, 12122

Offset: 1

R. Muller

Numbers written in the dyadic system [Smullyan, Stillwell]. - N. J. A. Sloane, Feb 13 2019
Logic-binary sequence: prefix it by the empty word to have all binary words on the alphabet {1,2}.
The least binary word of length k is a(2^k - 1).
See Mathematica program for logic-binary sequence using (0,1) in place of (1,2); the sequence starts with 0,1,00,01,10. - Clark Kimberling, Feb 09 2012
A007953(a(n)) = A014701(n+1); A007954(a(n)) = A048896(n). - Reinhard Zumkeller, Oct 26 2012
a(n) is n written in base 2 where zeros are not allowed but twos are. The two distinct digits used are 1, 2 instead of 0, 1. To obtain this sequence from the "canonical" base 2 sequence with zeros allowed, just replace any 0 with a 2 and then subtract one from the group of digits situated on the left: (10-->2; 100-->12; 110-->22; 1000-->112; 1010-->122). - Robin Garcia, Jan 31 2014
For numbers made of only two different digits, see also A007088 (digits 0 & 1), 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). Numbers with exactly two distinct (but unspecified) digits in base 10 are listed in A031955, for other bases in A031948-A031954. - M. F. Hasler, Apr 04 2015
The variant with digits {0, 1} instead of {1, 2} is obtained by deleting all initial digits in sequence A007088 (numbers written in base 2). - M. F. Hasler, Nov 03 2020

			Positive numbers may not start with 0 in the OEIS, otherwise this sequence would have been written as: 0, 1, 00, 01, 10, 11, 000, 001, 010, 011, 100, 101, 110, 111, 0000, 0001, 0010, 0011, 0100, 0101, 0110, 0111, 1000, 1001, 1010, 1011, 1100, 1101, 1110, 1111, 00000, 00001, 00010, 00011, 00100, 00101, 00110, 00111, 01000, 01001, 01010, 01011, ...
From _Hieronymus Fischer_, Jun 06 2012: (Start)
a(10)   = 122.
a(100)  = 211212.
a(10^3) = 222212112.
a(10^4) = 1122211121112.
a(10^5) = 2111122121211112.
a(10^6) = 2221211112112111112.
a(10^7) = 11221112112122121111112.
a(10^8) = 12222212122221111211111112.
a(10^9) = 22122211221212211212111111112. (End)

J.-P. Allouche and J. Shallit, Automatic Sequences, Cambridge Univ. Press, 2003, p. 2. - From N. J. A. Sloane, Jul 26 2012
K. Atanassov, On the 97th, 98th and the 99th Smarandache Problems, Notes on Number Theory and Discrete Mathematics, Sophia, Bulgaria, Vol. 5 (1999), No. 3, 89-93.
R. M. Smullyan, Theory of Formal Systems, Princeton, 1961.
John Stillwell, Reverse Mathematics, Princeton, 2018. See p. 90.

Hieronymus Fischer, Table of n, a(n) for n = 1..10000 (terms up to 2^10-2 from T. D. Noe, corrected by Sean A. Irvine, April 18 2019)
K. Atanassov, On Some of Smarandache's Problems, American Research Press, 1999, 16-21.
EMIS, Mirror site for Southwest Journal of Pure and Applied Mathematics
R. R. Forslund, A logical alternative to the existing positional number system, Southwest Journal of Pure and Applied Mathematics, Vol. 1, 1995.
R. R. Forslund, Positive Integer Pages
James E. Foster, A Number System without a Zero-Symbol, Mathematics Magazine, Vol. 21, No. 1. (1947), pp. 39-41.
F. Smarandache, Only Problems, Not Solutions!.
Index entries for 10-automatic sequences.

Cf. A007932 (digits 1-3), A059893, A045670, A052382 (digits 1-9), A059939, A059941, A059943, A032924, A084544, A084545, A046034 (prime digits 2,3,5,7), A089581, A084984 (no prime digits); A001742, A001743, A001744: loops; A202267 (digits 0, 1 and primes), A202268 (digits 1,4,6,8,9), A014261 (odd digits), A014263 (even digits).
Cf. A007088 (digits 0 & 1), 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).
Cf. A020450 (primes).

a007931 n = f (n + 1) where
   f x = if x < 2 then 0 else (10 * f x') + m + 1
     where (x', m) = divMod x 2
-- Reinhard Zumkeller, Oct 26 2012

[n: n in [1..100000] | Set(Intseq(n)) subset {1,2}]; // Vincenzo Librandi, Aug 19 2016

# Maple program to produce the sequence:
a:= proc(n) local m, r, d; m, r:= n, 0;
      while m>0 do d:= irem(m, 2, 'm');
        if d=0 then d:=2; m:= m-1 fi;
        r:= d, r
      od; parse(cat(r))/10
    end:
seq(a(n), n=1..100);  # Alois P. Heinz, Aug 26 2016
# Maple program to invert this sequence: given a(n), it returns n. - N. J. A. Sloane, Jul 09 2012
invert7931:=proc(u)
local t1,t2,i;
t1:=convert(u,base,10);
[seq(t1[i]-1,i=1..nops(t1))];
[op(%),1];
t2:=convert(%,base,2,10);
add(t2[i]*10^(i-1),i=1..nops(t2))-1;
end;

f[n_] := FromDigits[Rest@IntegerDigits[n + 1, 2] + 1]; Array[f, 42] (* Robert G. Wilson v Sep 14 2006 *)
(* Next, A007931 using (0,1) instead of (1,2) *)
d[n_] := FromDigits[Rest@IntegerDigits[n + 1, 2] + 1]; Array[FromCharacterCode[ToCharacterCode[ToString[d[#]]] - 1] &, 100] (* Peter J. C. Moses, at request of Clark Kimberling, Feb 09 2012 *)
Flatten[Table[FromDigits/@Tuples[{1,2},n],{n,5}]] (* Harvey P. Dale, Sep 13 2014 *)

apply( {A007931(n)=fromdigits([d+1|d<-binary(n+1)[^1]])}, [1..44]) \\ M. F. Hasler, Nov 03 2020, replacing older code from Mar 26 2015

/* inverse function */ apply( {A007931_inv(N)=fromdigits([d-1|d<-digits(N)],2)+2<M. F. Hasler, Nov 09 2020

def a(n): return int(bin(n+1)[3:].replace('1', '2').replace('0', '1'))
print([a(n) for n in range(1, 45)]) # Michael S. Branicky, May 13 2021

def A007931(n): return int(s:=bin(n+1)[3:])+(10**(len(s))-1)//9 # Chai Wah Wu, Jun 13 2025

To get a(n), write n+1 in base 2, remove initial 1, add 1 to all remaining digits: e.g., eleven (11) in base 2 is 1011; remove initial 1 and add 1 to remaining digits: a(10)=122. - Clark Kimberling, Mar 11 2003
Conversely, given a(n), to get n: subtract 1 from all digits, prefix with an initial 1, convert this binary number to base 10, subtract 1. E.g., a(6)=22 -> 11 -> 111 -> 7 -> 6. - N. J. A. Sloane, Jul 09 2012
a(n) = A053645(n+1)+A002275(A000523(n)) = a(n-2^b(n))+10^b(n) where b(n) = A059939(n) = floor(log_2(n+1)-1). - Henry Bottomley, Feb 14 2001
From Hieronymus Fischer, Jun 06 2012 and Jun 08 2012: (Start)
The formulas are designed to calculate base-10 numbers only using the digits 1 and 2.
a(n) = Sum_{j=0..m-1} (1 + b(j) mod 2)*10^j, where m = floor(log_2(n+1)), b(j) = floor((n+1-2^m)/(2^j)).
Special values:
a(k*(2^n-1)) = k*(10^n-1)/9, k= 1,2.
a(3*2^n-2) = (11*10^n-2)/9 = 10^n+2*(10^n-1)/9.
a(2^n-2) = 2*(10^(n-1)-1)/9, n>1.
Inequalities:
a(n) <= (10^log_2(n+1)-1)/9, equality holds for n=2^k-1, k>0.
a(n) > (2/10)*(10^log_2(n+1)-1)/9.
Lower and upper limits:
lim inf a(n)/10^log_2(n) = 1/45, for n --> infinity.
lim sup a(n)/10^log_2(n) = 1/9, for n --> infinity.
G.f.: g(x) = (1/(x(1-x)))*sum_{j=0..infinity} 10^j* x^(2*2^j)*(1 + 2 x^2^j)/(1 + x^2^j).
Also: g(x) = (1/(1-x))*(h_(2,0)(x) + h_(2,1)(x) - 2*h_(2,2)(x)), where h_(2,k)(x) = sum_{j>=0} 10^j*x^(2^(j+1)-1)*x^(k*2^j)/(1-x^2^(j+1)).
Also: g(x) = (1/(1-x)) sum_{j>=0} (1 - 3(x^2^j)^2 + 2(x^2^j)^3)*x^2^j*f_j(x)/(1-x^2^j), where f_j(x) = 10^j*x^(2^j-1)/(1-(x^2^j)^2). The f_j obey the recurrence f_0(x) = 1/(1-x^2), f_(j+1)(x) = 10x*f_j(x^2). (End)

Some crossrefs added by Hieronymus Fischer, Jun 06 2012

Edited by M. F. Hasler, Mar 26 2015

A036302 Composite numbers k such that the digits of the prime factors of k are either 1 or 2.

Original entry on oeis.org

4, 8, 16, 22, 32, 44, 64, 88, 121, 128, 176, 242, 256, 352, 422, 484, 512, 704, 844, 968, 1024, 1331, 1408, 1688, 1936, 2048, 2321, 2662, 2816, 3376, 3872, 4096, 4222, 4442, 4642, 5324, 5632, 6752, 7744, 8192, 8444, 8884, 9284, 10648, 11264, 13504, 14641, 15488, 16384

Offset: 1

Patrick De Geest, Dec 15 1998

All terms are a product of at least two terms of A020450. - Michel Marcus, Oct 02 2020

			422 = 2 * 211 is in the sequence as the digits of its prime factors 2 and 211 are either 1 or 2. - _David A. Corneth_, Sep 26 2020

David A. Corneth, Table of n, a(n) for n = 1..10000 (first 1000 terms from Alois P. Heinz)
Index entries sequences related to prime factors

Cf. A003596 (a subsequence), A020450, A036303-A036325.

[k:k in [2..15000]|  not IsPrime(k) and forall{a: a in PrimeDivisors(k)|Intseq(a) subset {1,2}}]; // Marius A. Burtea, Oct 08 2019

Select[Range[2,14650],!PrimeQ[#] && Complement[Flatten[IntegerDigits[First/@FactorInteger[#]]],{1,2}]=={} &] (* Jayanta Basu, May 25 2013 *)

Sum_{n>=1} 1/a(n) = Product_{p in A020450} p/(p-1) - Sum_{p in A020450} 1/p - 1 = 0.616325... - Amiram Eldar, Oct 14 2020

A036953 Primes having only {0, 1, 2} as digits.

Original entry on oeis.org

2, 11, 101, 211, 1021, 1201, 2011, 2111, 2221, 10111, 10211, 12011, 12101, 12211, 20011, 20021, 20101, 20201, 21001, 21011, 21101, 21121, 21211, 21221, 22111, 101021, 101111, 101221, 102001, 102101, 102121, 110221, 111121, 111211, 112111

Offset: 1

Patrick De Geest, Jan 04 1999

Number of n-digit terms d(n) = (1, 1, 2, 5, 16, 34, 76, 194, 543, 1469, 4094, 11017, ...); e.g., there are five 4-digit terms: 1021, 1201, 2011, 2111, 2221, hence d(4) = 5. - Zak Seidov, Jun 30 2013

Also, primes in A007089. - M. F. Hasler, Jul 25 2015

Zak Seidov, Table of n, a(n) for n = 1..10000
James Maynard and Brady Haran, Primes without a 7, Numberphile video (2019)
Index to entries for primes with digits in a given set

Subsequence of A107715.
Cf. A036952-A036964.
Cf. A020450 - A020472, A260044, A260267 - A260271, A199325 - A199329, A061247, A199340 - A199349, A217039, A079651.

Select[FromDigits/@Tuples[{0,1,2},6],PrimeQ] (* Harvey P. Dale, Jul 11 2017 *)

lista(n) = {forprime(p=2, n, if (vecmax(digits(p)) <= 2, print1(p, ", ")))} \\ Michel Marcus, Aug 02 2014

A036953={(n,show=0)->for(d=1,1e9,my(u=vector(d,i,10^(d-i))~);forvec(v=vector(d,i,if(i>1,if(iM. F. Hasler, Jul 25 2015

from gmpy2 import digits
from sympy import isprime
[int(digits(n,3)) for n in range(1000) if isprime(int(digits(n,3)))] # Chai Wah Wu, Jul 31 2014

Edited by M. F. Hasler, Jul 25 2015

A260267 Primes having only {1, 2, 4} as digits.

Original entry on oeis.org

2, 11, 41, 211, 241, 421, 2111, 2141, 2221, 2411, 2441, 4111, 4211, 4241, 4421, 4441, 11411, 12211, 12241, 12421, 14221, 14411, 21121, 21211, 21221, 22111, 22441, 24121, 24421, 41141, 41221, 41411, 42221, 44111, 44221, 111121, 111211, 112111, 112121, 112241

Offset: 1

Vincenzo Librandi, Jul 23 2015

A020450 and A020452 are subsequences.

All terms but the first one end with a digit "1". - M. F. Hasler, Jul 26 2015

Cf. similar sequences listed in A260266.

Cf. A020450, A020452.

[p: p in PrimesUpTo(4*10^5) | Set(Intseq(p)) subset [1, 4, 2]];

Select[Prime[Range[3 10^4]], Complement[IntegerDigits[#], {1, 4, 2}]=={} &]

A260267(n=50,show=0)={for(d=1,1e9,my(t,u=vector(d,i,10^(d-i))~);forvec(v=vector(d,i,[0,if(iM. F. Hasler, Jul 25 2015

A260266 Primes having only {0, 1, 4} as digits.

Original entry on oeis.org

11, 41, 101, 401, 4001, 4111, 4441, 10111, 10141, 11411, 14011, 14401, 14411, 40111, 41011, 41141, 41411, 44041, 44101, 44111, 100411, 101111, 101141, 101411, 110441, 114001, 114041, 140111, 140401, 140411, 141041, 141101, 400441, 401101, 401411, 404011

Offset: 1

Vincenzo Librandi, Jul 22 2015

A020449 and A020452 are subsequences.

All terms end with a digit "1". - M. F. Hasler, Jul 26 2015

Alois P. Heinz, Table of n, a(n) for n = 1..10000
Index to entries for primes with digits in a given set

Primes that contain only digits among {1,4,k}: this sequence (k=0), A260267 (k=2), A199341 (k=3), A260268 (k=5), A260269 (k=6), A079651 (k=7), A260270 (k=8), A260271 (k=9).
Cf. A020449, A020452.
Cf. A020450 - A020472, A036953, A260044, A260267 - A260271, A199325 - A199329, A061247, A199340 - A199349.

[p: p in PrimesUpTo(5*10^5) | Set(Intseq(p)) subset [1, 4, 0]];

Select[Prime[Range[4 10^4]], Complement[IntegerDigits[#], {1, 4, 0}]=={} &]

A260266(n=50,show=0)={for(d=1,1e9,my(t,u=vector(d,i,10^(d-i))~);forvec(v=vector(d,i,[i==1||i==d,1+(iM. F. Hasler, Jul 25 2015

A260889 Primes having only {1, 2, 7} as digits.

Original entry on oeis.org

2, 7, 11, 17, 71, 127, 211, 227, 271, 277, 727, 1117, 1171, 1217, 1277, 1721, 1777, 2111, 2221, 2711, 2777, 7121, 7127, 7177, 7211, 7717, 7727, 11117, 11171, 11177, 11717, 11777, 12211, 12227, 12277, 12721, 17117, 21121, 21211, 21221, 21227, 21277, 21727

Offset: 1

Vincenzo Librandi, Aug 04 2015

A020450, A020455 and A020459 are subsequences.

Alois P. Heinz, Table of n, a(n) for n = 1..10000
Index to entries for primes with digits in a given set

Cf. Primes that contain only the digits (k,1,7): A199327 (k=0), this sequence (k=2), A260379 (k=3), A079651 (k=4), A260828 (k=5), A260891 (k=6), A260892 (k=8), A260893 (k=9).

Cf. A020450, A020455, A020459.

[p: p in PrimesUpTo(3*10^4) | Set(Intseq(p)) subset [1, 2, 7]];

Select[Prime[Range[2 10^5]], Complement[IntegerDigits[#], {1, 2, 7}] == {} &]
Table[Select[FromDigits/@Tuples[{1,2,7},n],PrimeQ],{n,5}]//Flatten (* Harvey P. Dale, Apr 12 2018 *)

A385774 Primes having only {1, 2, 6} as digits.

Original entry on oeis.org

2, 11, 61, 211, 661, 1621, 2111, 2161, 2221, 2621, 6121, 6211, 6221, 6661, 11161, 11261, 11621, 12161, 12211, 12611, 16111, 16661, 21121, 21211, 21221, 21611, 21661, 22111, 22621, 26111, 26161, 26261, 61121, 61211, 61261, 66161, 66221, 111121, 111211

Offset: 1

Jason Bard, Jul 09 2025

Supersequence of A020450, A020454.

Cf. A000040, A385776.

[p: p in PrimesUpTo(10^6) | Set(Intseq(p)) subset [1, 2, 6]];

Flatten[Table[Select[FromDigits /@ Tuples[{1, 2, 6}, n], PrimeQ], {n, 7}]]

primes_with(, 1, [1, 2, 6]) \\ uses function in A385776

print(list(islice(primes_with("126"), 41))) # uses function/imports in A385776

A385775 Primes having only {1, 2, 8} as digits.

Original entry on oeis.org

2, 11, 181, 211, 281, 811, 821, 881, 1181, 1811, 2111, 2221, 2281, 8111, 8221, 8821, 11821, 12211, 12281, 12821, 18121, 18181, 18211, 21121, 21211, 21221, 21821, 21881, 22111, 22811, 28111, 28181, 28211, 81181, 81281, 82811, 88211, 88811, 111121

Offset: 1

Jason Bard, Jul 09 2025

Supersequence of A020450, A020456.

Cf. A000040, A385776.

[p: p in PrimesUpTo(10^6) | Set(Intseq(p)) subset [1, 2, 8]];

Flatten[Table[Select[FromDigits /@ Tuples[{1, 2, 8}, n], PrimeQ], {n, 7}]]

primes_with(, 1, [1, 2, 8]) \\ uses function in A385776

print(list(islice(primes_with("128"), 41))) # uses function/imports in A385776

A111488 Primes having only {0, 1, 3, 6} as digits.

Original entry on oeis.org

3, 11, 13, 31, 61, 101, 103, 113, 131, 163, 311, 313, 331, 601, 613, 631, 661, 1013, 1031, 1033, 1061, 1063, 1103, 1163, 1301, 1303, 1361, 1601, 1613, 1663, 3001, 3011, 3061, 3163, 3301, 3313, 3331, 3361, 3613, 3631, 6011, 6101, 6113, 6131, 6133, 6163

Offset: 1

Jonathan Vos Post, Nov 15 2005

Includes all repunit primes (A004022). Conjecture: an infinite sequence. Note twin primes: (11, 13), (101, 103), (311, 313), (1031, 1033), (1061, 1063), (1301, 1303), (6131, 6133), (10301, 10303), (10331, 10333), (13001, 13003).
In other words, primes with digits in the set {0,1,3,6}. - M. F. Hasler, Jul 25 2015
The number of 1's in the representation must be either 1 or 2 (mod 3), because otherwise the number would be divisible by 3 (and therefore composite). The only exception is the 3 itself. This excludes basically members of A038603. - R. J. Mathar, Jul 25 2015

Robert Israel, Table of n, a(n) for n = 1..10000
Index to entries for primes with digits in a given set

Cf. A000040, A000217, A004022, A038603.

Cf. also A020450 - A020472, A036953, A260044, A260267 - A260271, A199325 - A199329, A061247, A199340 - A199349.

f:= proc(x) local L,p;
  L:= subs([3=6,2=3],convert(x,base,4));
  p:= add(L[i]*10^(i-1),i=1..nops(L));
  if isprime(p) then p fi
end proc:
map(f, [$1..4^4]); # Robert Israel, Dec 18 2018

Select[Prime@ Range@ 1000, SubsetQ[{0, 1, 3, 6}, IntegerDigits@ #] &] (* Michael De Vlieger, Jul 25 2015 *)

A111488={(n, show=0, L=[0,1,3,6])->my(t); for(d=1,1e9,u=vector(d, i, 10^(d-i))~; forvec(v=vector(d,i,[1+(i==1&&!L[1]), #L]), ispseudoprime(t=vector(d, i, L[v[i]])*u)||next; show&print1(t", "); n--||return(t)))} \\ M. F. Hasler, Jul 25 2015

Corrected by Ray Chandler, Nov 19 2005

Name changed by Sean A. Irvine, Jul 21 2025

cp's OEIS Frontend

A385776 Primes having only {1, 2, 9} as digits.

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A007931 Numbers that contain only 1's and 2's. Nonempty binary strings of length n in lexicographic order.

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A036302 Composite numbers k such that the digits of the prime factors of k are either 1 or 2.

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A036953 Primes having only {0, 1, 2} as digits.

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A260267 Primes having only {1, 2, 4} as digits.

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A260266 Primes having only {0, 1, 4} as digits.

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A260889 Primes having only {1, 2, 7} as digits.