A020449 -id:A020449 - cp's OEIS frontend

A062339 Primes whose sum of digits is 4.

13, 31, 103, 211, 1021, 1201, 2011, 3001, 10111, 20011, 20101, 21001, 100003, 102001, 1000003, 1011001, 1020001, 1100101, 2100001, 10010101, 10100011, 20001001, 30000001, 101001001, 200001001, 1000000021, 1000001011, 1000010101, 1000020001, 1000200001, 1002000001, 1010000011

Offset: 1

Amarnath Murthy, Jun 21 2001

Is this sequence (and its brothers A062337, A062341 and A062343) infinite?

10^A049054(m)+3 and 3*10^A056807(m)+1 are subsequences. A107715 (primes containing only digits from set {0,1,2,3}) is a supersequence. Terms not containing the digit 3 are either terms of A020449 (primes that contain digits 0 and 1 only) or of A106100 (primes with maximal digit 2) - and thus terms of these sequences' union A036953 (primes containing only digits from set {0,1,2}). - Rick L. Shepherd, May 23 2005

			3001 is a prime with sum of digits = 4, hence belongs to the sequence.

T. D. Noe and Robert Israel, Table of n, a(n) for n = 1..10000 (n = 1..1000 from T. D. Noe)
Amin Witno, Numbers which factor as their digital sum times a prime, International Journal of Open Problems in Computer Science and Mathematics 3:2 (2010), pp. 132-136.

Subsequence of A062338, A107288, and A107715 (primes with digits <= 3).
A159352 is a subsequence.
Cf. A000040 (primes), A052218 (digit sum = 4), A061239 (primes == 4 (mod 9)).
Cf. Primes p with digital sum equal to k: {2, 11 and 101} for k=2; this sequence (k=4), A062341 (k=5), A062337 (k=7), A062343 (k=8), A107579 (k=10), A106754 (k=11), A106755 (k=13), A106756 (k=14), A106757 (k=16), A106758 (k=17), A106759 (k=19), A106760 (k=20), A106761 (k=22), A106762 (k=23), A106763 (k=25), A106764 (k=26), A048517 (k=28), A106766 (k=29), A106767 (k=31), A106768 (k=32), A106769 (k=34), A106770 (k=35), A106771 (k=37), A106772 (k=38), A106773 (k=40), A106774 (k=41), A106775 (k=43), A106776 (k=44), A106777 (k=46), A106778 (k=47), A106779 (k=49), A106780 (k=50), A106781 (k=52), A106782 (k=53), A106783 (k=55), A106784 (k=56), A106785 (k=58), A106786 (k=59), A106787 (k=61), A107617 (k=62), A107618 (k=64), A107619 (k=65), A106807 (k=67), A244918 (k=68), A181321 (k=70).
Cf. A049054 (10^k+3 is prime), A159352 (these primes).
Cf. A056807 (3*10^k+1 is prime), A259866 (these primes).
Cf. A020449 (primes with digits 0 and 1), A036953 (primes with digits <= 2), A106100 (primes with largest digit = 2), A069663, A069664 (smallest resp. largest n-digit prime with minimum digit sum).

[p: p in PrimesUpTo(800000000) | &+Intseq(p) eq 4]; // Vincenzo Librandi, Jul 08 2014

N:= 20: # to get all terms < 10^N
B[1]:= {1}:
B[2]:= {2}:
B[3]:= {3}:
A:= {}:
for d from 2 to N do
   B[4]:= map(t -> 10*t+1,B[3]) union  map(t -> 10*t+3,B[1]);
   B[3]:= map(t -> 10*t, B[3]) union map(t -> 10*t+1,B[2]) union map(t -> 10*t+2,B[1]);
   B[2]:= map(t -> 10*t, B[2]) union map(t -> 10*t+1,B[1]);
   B[1]:= map(t -> 10*t, B[1]);
   A:= A union select(isprime,B[4]);
od:
sort(convert(A,list)); # Robert Israel, Dec 28 2015

Union[FromDigits/@Select[Flatten[Table[Tuples[{0,1,2,3},k],{k,9}],1],PrimeQ[FromDigits[#]]&&Total[#]==4&]] (* Jayanta Basu, May 19 2013 *)
FromDigits/@Select[Tuples[{0,1,2,3},10],Total[#]==4&&PrimeQ[FromDigits[#]]&] (* Harvey P. Dale, Jul 23 2025 *)

for(a=1,20,for(b=0,a,for(c=0,b,if(isprime(k=10^a+10^b+10^c+1),print1(k", "))))) \\ Charles R Greathouse IV, Jul 26 2011

select( {is_A062339(p,s=4)=sumdigits(p)==s&&isprime(p)}, primes([1,10^7])) \\ 2nd optional parameter for similar sequences with other digit sums. M. F. Hasler, Mar 09 2022

A062339_upto_length(L,s=4,a=List(),u=[10^(L-k)|k<-[1..L]])=forvec(d=[[1,L]|i<-[1..s]], isprime(p=vecsum(vecextract(u,d))) && listput(a,p),1); Vecrev(a) \\ M. F. Hasler, Mar 09 2022

Intersection of A052218 (digit sum 4) and A000040 (primes). - M. F. Hasler, Mar 09 2022

Corrected and extended by Larry Reeves (larryr(AT)acm.org), Jul 06 2001
More terms from Rick L. Shepherd, May 23 2005
More terms from Lekraj Beedassy, Dec 19 2007

A235639 Primes whose base-9 representation is also the base-6 representation of a prime.

Original entry on oeis.org

2, 3, 5, 19, 23, 41, 113, 127, 131, 163, 199, 271, 419, 433, 739, 743, 761, 919, 991, 1009, 1013, 1063, 1153, 1171, 1459, 1481, 1499, 1553, 1567, 1571, 1733, 1747, 1783, 1873, 1913, 2237, 2377, 2381, 2539, 2557, 2593, 2633, 2939, 3011, 3079, 3083, 3187, 3259, 3331, 3659

Offset: 1

M. F. Hasler, Jan 13 2014

This sequence is part of the two-dimensional array of sequences based on this same idea for any two different bases b, c > 1. Sequence A235265 and A235266 are the most elementary ones in this list. Sequences A089971, A089981 and A090707 through A090721, and sequences A065720 - A065727, follow the same idea with one base equal to 10.

			19 = 21_9 and 21_6 = 13 are both prime, so 19 is a term.
509 = 625_9 and 625_6 = 17 are both prime, but 625 is not a valid base-6 integer, so 509 is not a term.

Robert Israel, Table of n, a(n) for n = 1..10000
M. F. Hasler, Primes whose base c expansion is also the base b expansion of a prime

Cf. A231481, A235265, A235266, A152079, A235461 - A235482, A065720 - A065727, A235394, A235395, A089971 ⊂ A020449, A089981, A090707 - A091924, A235615 - A235638. See the LINK for further cross-references.

R:= 2: x:= 2: count:= 1:
while count < 100 do
  x:= nextprime(x);
  L:= convert(x,base,6);
  y:= add(9^(i-1)*L[i],i=1..nops(L));
  if isprime(y) then count:= count+1; R:= R, y fi
od:
R; # Robert Israel, May 18 2020

is(p,b=6,c=9)=vecmax(d=digits(p,c))

forprime(p=1,3e3,is(p,9,6)&&print1(vector(#d=digits(p,6),i,9^(#d-i))*d~,",")) \\ To produce the terms, this is more efficient than to select them using straightforwardly is(.)=is(.,6,9)

A199340 Primes having only {0, 3, 4} as digits.

Original entry on oeis.org

3, 43, 433, 443, 3343, 3433, 4003, 30403, 33343, 33403, 34033, 34303, 34403, 40343, 40433, 43003, 43403, 300043, 300343, 304033, 304303, 304433, 330433, 333433, 334043, 334333, 334403, 343303, 343333, 343433, 400033, 403003, 403043, 403433, 430303, 430333

Offset: 1

M. F. Hasler, Nov 05 2011

All terms end in '3'. This could be used to speed up the given program.

A020461 is a subsequence. - Vincenzo Librandi, Jul 23 2015

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

Cf. Primes that contain only the digits (3,4,k): this sequence (k=0), A199341 (k=1), A199342 (k=2), A199345 (k=5), A199346 (k=6), A199347 (k=7), A199348 (k=8), A199349 (k=9).
Cf. A020449 - A020472, A199325 - A199329.
Cf. A058429, A058430.

[p: p in PrimesUpTo(5*10^5) | Set(Intseq(p)) subset [3, 4, 0]]; // Vincenzo Librandi, Jul 23 2015

Select[Prime[Range[5 10^4]], Complement[IntegerDigits[#], {3, 4, 0}]=={} &] (* Vincenzo Librandi, Jul 23 2015 *)
Select[FromDigits/@Tuples[{0,3,4},6],PrimeQ] (* Harvey P. Dale, Mar 21 2020 *)
Select[10#+3&/@FromDigits/@Tuples[{0,3,4},5],PrimeQ] (* Harvey P. Dale, May 02 2022 *)

a(n, list=0, L=[0, 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)))} \\ Syntax updated for current PARI version. - M. F. Hasler, Jul 25 2015

{forprime(p=3,1e6,p%10==3&&!setminus(Set(digits(p)),[3,4])&&print1(p","))} \\ [0] evaluates to false. - M. F. Hasler, Jul 25 2015

A061247 Primes having only {0, 1, 8} as digits.

Original entry on oeis.org

11, 101, 181, 811, 881, 1181, 1801, 1811, 8011, 8081, 8101, 8111, 10111, 10181, 11801, 18181, 80111, 81001, 81101, 81181, 88001, 88801, 88811, 100801, 100811, 101081, 101111, 108011, 108881, 110881, 118081, 118801, 180001, 180181, 180811

Offset: 1

Amarnath Murthy, Apr 23 2001

The intersection with A007500 is listed in A199328. - M. F. Hasler, Nov 05 2011

			a(6) = 1801, 1801 is a prime and consists of only 1, 8 and 0.

Robert Israel, Table of n, a(n) for n = 1..17482

Cf. A061246, A020449-A020472, A199325-A199329.

[NthPrime(n): n in [1..2*10^4] | forall{d: d in Intseq(NthPrime(n)) | d in [0, 1, 8]}]; // Vincenzo Librandi, May 15 2019

N:= 1000: # to get the first N entries
count:= 0:
allowed:= {0,1,8}:
nallowed:= nops(allowed):
subst:= seq(i=allowed[i+1],i=0..nallowed-1);
for d from 1 while count < N do
  for x1 from 1 to nallowed-1 while count < N do
    for t from 0 to nallowed^d-1  while count < N do
      L:= subs(subst,convert(x1*nallowed^d+t,base,nallowed));
      X:= add(L[i]*10^(i-1),i=1..d+1);
      if isprime(X) then
          count:= count+1;
          A[count]:= X;
      fi
od od od:
seq(A[n],n=1..N); # Robert Israel, Apr 20 2014

Select[Prime[Range[50000]],Length[Union[{0,1,8},IntegerDigits[ # ]]] == 3&] (* Stefan Steinerberger, Jun 10 2007 *)
Select[FromDigits/@Tuples[{0,1,8},6],PrimeQ] (* Harvey P. Dale, Jan 12 2016 *)

a(n=50, L=[0, 1, 8], show=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]), ispseudoprime(t=vector(d, i, L[v[i]])*u) || next; show && print1(t", "); n-- || return(t)))} \\ M. F. Hasler, Nov 05 2011

Corrected and extended by Stefan Steinerberger, Jun 10 2007

A199349 Primes having only {3, 4, 9} as digits.

Original entry on oeis.org

3, 43, 349, 433, 439, 443, 449, 499, 3343, 3433, 3449, 3499, 3943, 4339, 4349, 4493, 4933, 4943, 4993, 4999, 9343, 9349, 9433, 9439, 9949, 33343, 33349, 33493, 34439, 34499, 34939, 34949, 39343, 39439, 39443, 39499, 43399, 43499, 43933, 43943, 44449, 44939, 49333, 49339, 49393, 49433, 49499, 49939, 49943, 49993

Offset: 1

M. F. Hasler, Nov 05 2011

A020461 and A020466 are subsequences. - Vincenzo Librandi, Jul 30 2015

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

Cf. Primes that contain only the digits (3,4,k): A199340 (k=0), A199341 (k=1), A199342 (k=2), A199345 (k=5), A199346 (k=6), A199347 (k=7), A199348 (k=8).

Cf. A020449 - A020472, A199325 - A199329.

[p: p in PrimesUpTo(2*10^5) | Set(Intseq(p)) subset [3, 4, 9]]; // Vincenzo Librandi, Jul 30 2015

Select[Prime[Range[2 10^4]], Complement[IntegerDigits[#], {3, 4, 9}]=={} &] (* Vincenzo Librandi, Jul 30 2015 *)
Select[Flatten[Table[FromDigits/@Tuples[{3,4,9},n],{n,5}]],PrimeQ] (* Harvey P. Dale, May 02 2023 *)

a(n, list=0, L=[3,4,9], 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=vecextract(L,v)*u) || next; reqpal && !isprime(A004086(t)) && next; list && print1(t", "); n--||return(t)))}

A020471 Primes that contain digits 7 and 9 only.

Original entry on oeis.org

7, 79, 97, 797, 977, 997, 77797, 77977, 77999, 79777, 79979, 79997, 79999, 97777, 777977, 777979, 779797, 797977, 799979, 799999, 999979, 7777997, 7779979, 7779997, 7797799, 7797997, 7799797, 7799999, 7977779, 7977797, 7977799, 7977997

Offset: 1

David W. Wilson

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

Subsequence of A030096.

Cf. A020449 (digits 0 & 1), ..., A020472 (digits 8 & 9).

[p: p in PrimesUpTo(7977997) | Set(Intseq(p)) subset [7,9]]; // Vincenzo Librandi, Jul 28 2012

Flatten[Table[Select[FromDigits/@Tuples[{7,9},n],PrimeQ],{n,7}]] (* Vincenzo Librandi, Jul 28 2012 *)

Edited by M. F. Hasler, Jul 26 2015

A020450 Primes that contain digits 1 and 2 only.

Original entry on oeis.org

2, 11, 211, 2111, 2221, 12211, 21121, 21211, 21221, 22111, 111121, 111211, 112111, 112121, 1111211, 1121221, 1212121, 1212221, 1221221, 2121121, 2211211, 2221111, 11221211, 12111221, 12121121, 12121211, 12122111, 12122221, 12212111, 12222121

Offset: 1

David W. Wilson

Jason Bard, Table of n, a(n) for n = 1..10000 (first 1000 terms from Vincenzo Librandi)

Cf. A020449 (digits 0 & 1), ..., A020472 (digits 8 & 9). [From M. F. Hasler, Mar 18 2010]

Subsequence of A007931.

[p: p in PrimesUpTo(12222121) | Set(Intseq(p)) subset [1, 2]]; // Vincenzo Librandi, Jul 28 2012

Flatten[Table[Select[FromDigits/@Tuples[{1,2},n],PrimeQ],{n,8}]] (* Vincenzo Librandi, Jul 28 2012 *)

for(nd=1,9, forvec(v=vector(nd,i,[49,50-(i==nd && i>1)]), isprime(t=eval(Strchr(Vecsmall(v)))) && print1(t","))) \\ M. F. Hasler, Mar 18 2010

from sympy import primerange
def checkd(a, c):
    b =  set(int(i) for i in set(str(a)))
    return b.issubset(c)
for n in primerange(2, 2000000):
    if checkd(n, [1, 2]):
        print(n)
# Abhiram R Devesh, May 08 2015

A020469 Primes that contain digits 6 and 7 only.

Original entry on oeis.org

7, 67, 677, 67777, 76667, 76777, 666667, 677767, 767677, 777677, 6676667, 6676777, 6677677, 6677767, 6677777, 6766667, 6766777, 6776677, 7666667, 7667677, 7667767, 7766767, 7766777, 7777667, 66666667, 66677777, 66776777, 67667777, 67766767, 67776677, 67776767

Offset: 1

David W. Wilson

Michael S. Branicky, Table of n, a(n) for n = 1..10000 (terms 1..1000 from Vincenzo Librandi)

Cf. A020449-A020472.

Flatten[Table[Select[FromDigits/@Tuples[{6,7},n],PrimeQ],{n,8}]] (* Vincenzo Librandi, Jul 27 2012 *)

from sympy import isprime
from itertools import count, islice, product
def agen(): # generator of terms
    yield 7
    for d in count(2):
        for first in product("67", repeat=d-1):
            t = int("".join(first) + "7")
            if isprime(t): yield t
print(list(islice(agen(), 31))) # Michael S. Branicky, Nov 15 2022

A077718 Primes which can be expressed as sum of distinct powers of 4.

Original entry on oeis.org

5, 17, 257, 277, 337, 1093, 1109, 1297, 1301, 1361, 4177, 4357, 4373, 4421, 5189, 5381, 5393, 5441, 16453, 16657, 16661, 17477, 17489, 17669, 17681, 17729, 17749, 20549, 20753, 21521, 21569, 21589, 21841, 65537, 65557, 65617, 65809, 66629

Offset: 1

Amarnath Murthy, Nov 19 2002

nonn

Primes whose base 4 representation contains only zeros and 1's.

As a subsequence of primes in A000695, these could be called Moser-de Bruijn primes. See also A235461 for those terms whose base 4 representation also represents a prime in base 2. - M. F. Hasler, Jan 11 2014

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

Cf. A020449, A000695, A077717, A077719, A077720, A077721, A077722.

f:= proc(n) local L,x;
  L:= convert(n,base,2);
  x:= 1+add(L[i]*4^i,i=1..nops(L));
  if isprime(x) then x fi
end proc:
map(f, [$1..1000]); # Robert Israel, Sep 06 2018

Select[Prime[Range[6650]],Max[IntegerDigits[#,4]]<=1&] (* Jayanta Basu, May 22 2013 *)

for(i=1,999,isprime(b=vector(#b=binary(i),j,4^(#b-j))*b~)&&print1(b",")) \\ - M. F. Hasler, Jan 12 2014

More terms from Sascha Kurz, Jan 03 2003

A077722 Primes which can be expressed as sums of distinct powers of 8.

Original entry on oeis.org

73, 521, 577, 4673, 32833, 33289, 33353, 36929, 37441, 262153, 262217, 262657, 295433, 299017, 299521, 2097673, 2101249, 2101313, 2134529, 2359369, 2359873, 2363393, 2363401, 2392073, 16777289, 16777729, 16810049, 16810561, 16814089

Offset: 1

Amarnath Murthy, Nov 19 2002

nonn

Primes whose base 8 representations contain only 0's and 1's.

Intersection of A000040 and A033045. - Michel Marcus, Sep 14 2013

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000

Cf. A020449, A000695, A033045, A077717, A077718, A077719, A077720, A077721, A077723.

isok(n) = {digs = digits(n, 8); for (i = 1, #digs, if (digs[i] > 1, return (0));); return (1);}
lista(nn) = {forprime (p=1, nn, if (isok(p), print1(p, ", ");););} \\ Michel Marcus, Sep 14 2013

forstep(n=7,999,2,t=fromdigits(binary(n),8); if(isprime(t), print1(t", "))) \\ Charles R Greathouse IV, Jun 08 2015

More terms from Francois Jooste (phukraut(AT)hotmail.com), Dec 23 2002

cp's OEIS Frontend

A062339 Primes whose sum of digits is 4.

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A235639 Primes whose base-9 representation is also the base-6 representation of a prime.

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A199340 Primes having only {0, 3, 4} as digits.

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A061247 Primes having only {0, 1, 8} as digits.

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A199349 Primes having only {3, 4, 9} as digits.

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A020471 Primes that contain digits 7 and 9 only.

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A020450 Primes that contain digits 1 and 2 only.

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