A020451 -id:A020451 - cp's OEIS frontend

A032917 Numbers having only digits 1 and 3 in their decimal representation.

1, 3, 11, 13, 31, 33, 111, 113, 131, 133, 311, 313, 331, 333, 1111, 1113, 1131, 1133, 1311, 1313, 1331, 1333, 3111, 3113, 3131, 3133, 3311, 3313, 3331, 3333, 11111, 11113, 11131, 11133, 11311, 11313, 11331, 11333, 13111, 13113

Offset: 1

Clark Kimberling

Numbers n such that product of digits of n is a power of 3. - Vincenzo Librandi Aug 19 2016

Vincenzo Librandi, Table of n, a(n) for n = 1..1000

CF. A020451 (primes).

[n: n in [1..14000] | Set(IntegerToSequence(n, 10)) subset {1, 3}]; // Vincenzo Librandi, Jun 02 2012

Flatten[Table[FromDigits[#,10]&/@Tuples[{1,3},n],{n,5}]] (* Vincenzo Librandi, Jun 02 2012 *)

for(n=1, 5, p=2*vector(n, i, 10^(n-i))~; forvec(d=vector(n, i, [1, 3]/2), print1(d*p,","))) \\ M. F. Hasler, Mar 10 2014

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

def A032917(n): return (int(bin(m:=n+1)[3:])<<1) + (10**(m.bit_length()-1)-1)//9 # Chai Wah Wu, Oct 13 2023

Definition reworded by M. F. Hasler, Mar 10 2014

A199341 Primes having only {1, 3, 4} as digits.

Original entry on oeis.org

3, 11, 13, 31, 41, 43, 113, 131, 311, 313, 331, 431, 433, 443, 1433, 3313, 3331, 3343, 3413, 3433, 4111, 4133, 4441, 11113, 11131, 11311, 11411, 11443, 13313, 13331, 13411, 13441, 14143, 14341, 14411, 14431, 31333, 33113, 33311, 33331, 33343, 33413, 34141, 34313

Offset: 1

M. F. Hasler, Nov 05 2011

A020451, A020452 and A020461 are subsequences. - Vincenzo Librandi, Jul 26 2015

Robert Israel, Table of n, a(n) for n = 1..10000
Andrew Granville, Missing digits, and good approximations, arXiv:2308.03126 [math.NT], 2023. See p. 4.

Cf. A020449 - A020472, A199325 - A199329.

Cf. similar sequences listed in A199340.

[p: p in PrimesUpTo(10^5) | Set(Intseq(p)) subset [3, 4, 1]]; // Vincenzo Librandi, Jul 26 2015

Dmax:= 5: # to get all terms < 10^Dmax
Cd:= {1,3,4}:
C:= Cd:
for d from 2 to Dmax do
  Cd:= map(t -> (10*t+1,10*t+3,10*t+4),Cd);
  C:= C union Cd;
od:
sort(convert(select(isprime,C),list)); # Robert Israel, Jul 26 2015

Select[Prime[Range[4 10^3]], Complement[IntegerDigits[#], {3, 4, 1}]=={} &] (* Vincenzo Librandi, Jul 26 2015 *)

a(n, list=0, L=[1, 3, 4], reqpal=0)={my(t); for(d=1, 1e9, u=vector(d, i, 10^(d-i))~; forvec(v=vector(d, i, [1+(i==1&!L[1]), #L]), isprime(t=vector(d, i, L[v[i]])*u)||next; reqpal & !isprime(A004086(t)) & next; list & print1(t", "); n--||return(t)))}

A260379 Primes having only {1, 3, 7} as digits.

Original entry on oeis.org

3, 7, 11, 13, 17, 31, 37, 71, 73, 113, 131, 137, 173, 311, 313, 317, 331, 337, 373, 733, 773, 1117, 1171, 1373, 1733, 1777, 3137, 3313, 3331, 3371, 3373, 3733, 7177, 7331, 7333, 7717, 11113, 11117, 11131, 11171, 11173, 11177, 11311, 11317, 11717, 11731, 11777

Offset: 1

Vincenzo Librandi, Jul 24 2015

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

Subsequence of A030096 and A155055. A020451, A020455, and A020463 are subsequences.
Cf. similar sequences listed in A260378.
Cf. A000040, A020451, A020455, A020463.

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

Select[Prime[Range[2 10^3]], Complement[IntegerDigits[#], {1, 3, 7}]=={} &]

A260224 Primes having only {1, 3, 5} as digits.

Original entry on oeis.org

3, 5, 11, 13, 31, 53, 113, 131, 151, 311, 313, 331, 353, 1151, 1153, 1511, 1531, 1553, 3313, 3331, 3511, 3533, 5113, 5153, 5333, 5351, 5531, 11113, 11131, 11311, 11351, 11353, 11551, 13151, 13313, 13331, 13513, 13553, 15131, 15313, 15331, 15511, 15551

Offset: 1

Vincenzo Librandi, Jul 21 2015

Jason Bard, Table of n, a(n) for n = 1..10000 (first 1000 terms from Chai Wah Wu)
Index to entries for primes with digits in a given set

Subsequence of A030096. A004022, A020451, A020453, and A020462 are subsequences.
Cf. similar sequences listed in A260223.
Cf. A020451, A020453, A020462, A385776.

[p: p in PrimesUpTo(40000) | Set(Intseq(p)) subset [3, 5, 1]];

Select[Prime[Range[3 10^3]], Complement[IntegerDigits[#], {3, 5, 1}]=={} &]
Select[Flatten[Table[FromDigits/@Tuples[{1,3,5},n],{n,5}]],PrimeQ] (* Harvey P. Dale, Mar 03 2020 *)

from gmpy2 import is_prime, mpz
from itertools import product
A260224_list = [int(''.join(x)) for n in range(1,10) for x in product('135',repeat=n) if is_prime(mpz(''.join(x)))] # Chai Wah Wu, Jul 21 2015

A385777 Primes having only {1, 3, 6} as digits.

Original entry on oeis.org

3, 11, 13, 31, 61, 113, 131, 163, 311, 313, 331, 613, 631, 661, 1163, 1361, 1613, 1663, 3163, 3313, 3331, 3361, 3613, 3631, 6113, 6131, 6133, 6163, 6311, 6361, 6661, 11113, 11131, 11161, 11311, 11633, 13163, 13313, 13331, 13613, 13633, 16111

Offset: 1

Jason Bard, Jul 09 2025

Supersequence of A020451, A020454.

Cf. A000040, A385776.

[p: p in PrimesUpTo(10^6) | Set(Intseq(p)) subset [1, 3, 6]];

Flatten[Table[Select[FromDigits /@ Tuples[{1, 3, 6}, n], PrimeQ], {n, 7}]]

primes_with(,1,[1,3,6]) \\ uses function in A385776

print(list(islice(primes_with("136"), 41))) # uses function/imports in A385776

A385778 Primes having only {1, 3, 8} as digits.

Original entry on oeis.org

3, 11, 13, 31, 83, 113, 131, 181, 311, 313, 331, 383, 811, 881, 883, 1181, 1381, 1811, 1831, 3181, 3313, 3331, 3833, 3881, 8111, 8311, 8831, 11113, 11131, 11311, 11383, 11813, 11831, 11833, 13183, 13313, 13331, 13381, 13831, 13883, 18131, 18133, 18181, 18311

Offset: 1

Jason Bard, Jul 13 2025

Supersequence of A020451, A020456, A020464.

Cf. A000040, A385776.

[p: p in PrimesUpTo(10^6) | Set(Intseq(p)) subset [1, 3, 8]];

Flatten[Table[Select[FromDigits /@ Tuples[{1, 3, 8}, n], PrimeQ], {n, 7}]]

primes_with(, 1, [1, 3, 8]) \\ uses function in A385776

print(list(islice(primes_with("138"), 41))) # uses function/imports in A385776

A036303 Composite numbers whose prime factors contain no digits other than 1 and 3.

Original entry on oeis.org

9, 27, 33, 39, 81, 93, 99, 117, 121, 143, 169, 243, 279, 297, 339, 341, 351, 363, 393, 403, 429, 507, 729, 837, 891, 933, 939, 961, 993, 1017, 1023, 1053, 1089, 1179, 1209, 1243, 1287, 1331, 1441, 1469, 1521, 1573, 1703, 1859, 2187, 2197, 2511, 2673, 2799

Offset: 1

Patrick De Geest, Dec 15 1998

All terms are a product of at least two terms of A020451. - David A. Corneth, Oct 09 2020

			The composite 117 = 3^2 * 13 is in the sequence as the digits of the prime factors are either 1 or 3. - _David A. Corneth_, Oct 17 2020

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

Cf. A003597, A020451, A036302-A036325.

Select[Range[3000],CompositeQ[#]&&SubsetQ[{1,3},Union[Flatten[IntegerDigits/@FactorInteger[#][[;;,1]]]]]&] (* Harvey P. Dale, Jan 08 2025 *)

from sympy import factorint
def ok(n):
    f = factorint(n)
    return sum(f.values()) > 1 and all(set(str(p)) <= set("13") for p in f)
print(list(filter(ok, range(2800)))) # Michael S. Branicky, Sep 27 2021

Sum_{n>=1} 1/a(n) = Product_{p in A020451} (p/(p - 1)) - Sum_{p in A020451} 1/p - 1 = 0.3374936085... . - Amiram Eldar, May 18 2022

A036930 Smallest n-digit prime containing only digits 1 and 3, or 0 if no such prime exists.

Original entry on oeis.org

3, 11, 113, 3313, 11113, 113111, 1111333, 11111131, 111111113, 1111113313, 11111111113, 111111133333, 1111111111333, 11111111113133, 111111111113113, 1111111111313131, 11111111111131333, 111111111111111131

Offset: 1

Patrick De Geest, Jan 04 1999

Jinyuan Wang, Table of n, a(n) for n = 1..500

Cf. A020451, A036229, A036303.

Flatten[Table[Select[FromDigits/@Tuples[{1,3},n],PrimeQ,1],{n,18}]] (* Harvey P. Dale, Jul 23 2012 *)

Corrected by Harvey P. Dale, Jul 23 2012

A335787 Emirps containing only the digits 1 and 3.

Original entry on oeis.org

13, 31, 113, 311, 113131, 131311, 1131131, 1133131, 1133333, 1311311, 1313311, 1331333, 1333313, 3111313, 3131113, 3133331, 3331331, 3333311, 11113111, 11131111, 11311133, 11313311, 11331311, 11333131, 13131133, 13133311, 31111313, 31311113, 33111311, 33113131

Offset: 1

Daniel Starodubtsev, Jun 23 2020

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

Intersection of A006567 and A032917.

Subsequence of A020451.

Select[Flatten[Table[FromDigits[#, 10] & /@ Tuples[{1, 3}, n], {n, 8}]], # != (r = IntegerReverse[#]) && PrimeQ[#] && PrimeQ[r] &] (* Amiram Eldar, Jun 23 2020 after Vincenzo Librandi at A032917 *)

A358020 Least prime number > prime(n) (n >= 5) whose set of decimal digits coincides with the set of decimal digits of prime(n), or -1 if no such prime exists.

Original entry on oeis.org

1111111111111111111, 31, 71, 191, 223, 229, 113, 73, 4111, 433, 4447, 353, 599, 661, 677, 1117, 337, 97, 383, 8999, 797, 10111, 1013, 701, 1009, 131, 271, 311, 173, 193, 419, 1151, 571, 613, 617, 317, 197, 811, 199, 1193, 719, 911, 2111, 233, 277, 929, 2333, 293, 421, 521, 2557

Offset: 5

Jean-Marc Rebert, Oct 24 2022

			prime(6) = 13 and the prime number 31 have the same set of digits {1,3}, and 31 is the smallest such number, hence a(6) = 31.
prime(13) = 41 and the prime number 4111 have the same set of digits {1,4}, and 4111 is the smallest such number, hence a(13) = 4111.
prime(20) = 71 and the prime number 1117 have the same set of digits {1,7}, and 1117 is the smallest such number, hence a(20) = 1117.

Jean-Marc Rebert, Table of n, a(n) for n = 5..10000

Cf. A000040, A357096, A004022, A004023, A020451, A020455, A050288, A166681.

N:= 60: # for a(5)..a(N)
A:= Array(5..N):
R:= 1111111111111111111:
A[5]:= R: count:= 1:
for k from 6 while count < N-4 do
  p:= ithprime(k);
  S:= convert(convert(p,base,10),set);
  if assigned(V[S]) and V[S]<=N then A[V[S]]:= p; count:=count+1;  fi;
  V[S]:= k;
od:
convert(A,list); # Robert Israel, Oct 25 2022

a(n)=my(m=Set(digits(prime(n)))); if(n<5, return()); if(n==5,return(1111111111111111111));forprime(p=prime(n+1), , if(Set(digits(p))==m, return(p)))

from sympy import isprime, prime
from itertools import count, product
def a(n):
    pn = prime(n)
    s = str(pn)
    for d in count(len(s)):
        for p in product(set(s), repeat=d):
            if p[0] == "0": continue
            t = int("".join(p))
            if t > pn and isprime(t):
                return t
print([a(n) for n in range(5, 56)]) # Michael S. Branicky, Oct 25 2022

cp's OEIS Frontend

A032917 Numbers having only digits 1 and 3 in their decimal representation.

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A199341 Primes having only {1, 3, 4} as digits.

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A260379 Primes having only {1, 3, 7} as digits.

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A260224 Primes having only {1, 3, 5} as digits.

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A385777 Primes having only {1, 3, 6} as digits.

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A385778 Primes having only {1, 3, 8} as digits.

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A036303 Composite numbers whose prime factors contain no digits other than 1 and 3.

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