A020472 -id:A020472 - cp's OEIS frontend

A199302 Palindromic primes in the sense of A007500 with digits '0', '1' and '2' only.

2, 11, 101, 1021, 1201, 110221, 111211, 112111, 120121, 121021, 122011, 1000211, 1010201, 1020101, 1022011, 1022201, 1101211, 1102111, 1102201, 1111021, 1112011, 1120001, 1120121, 1120211, 1121011, 1201021, 1201111, 1210211, 1212121, 1221221, 10002121

Offset: 1

M. F. Hasler, Nov 04 2011

All terms except for the initial 2 start and end in the digit 1.

Cf. A020449 - A020472, A199325 - A199329, A199303 - A199306.

[p: p in PrimesUpTo(10^8) | Set(Intseq(p)) subset [0..2] and IsPrime(Seqint(Reverse(Intseq(p))))];  // Bruno Berselli, Nov 07 2011

allow=Vec("012");forprime(p=1,default(primelimit),setminus( Set( Vec(Str( p ))),allow)&next;isprime(A004086(p))&print1(p",")) /* better use the much more efficient code below */

a(n=50,list=0,L=[0,1,2],needpal=1)={ 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; needpal & !isprime(A004086(t)) & next; list & print1(t","); n-- || return(t)))}  \\ M. F. Hasler, Nov 06 2011

from itertools import count, islice, product
from sympy import isprime
def A199302_gen(): return (n for n in (int(t+''.join(s)) for l in count(0) for t in '12' for s in product('012',repeat=l)) if isprime(n) and isprime(int(str(n)[::-1])))
A199302_list = list(islice(A199302_gen(),20)) # Chai Wah Wu, Jan 04 2022

A199306 Palindromic primes in the sense of A007500 with digits '0', '1' and '6' only.

Original entry on oeis.org

11, 101, 1061, 1601, 10061, 10601, 11161, 16001, 16061, 16111, 16661, 101611, 106661, 116101, 166601, 1011601, 1016011, 1016611, 1061101, 1066111, 1106101, 1110611, 1111661, 1116601, 1160111, 1160611, 1166101, 1600061, 1611161, 1616161, 1660661, 1661111, 10011101, 10100161, 10106111

Offset: 1

M. F. Hasler, Nov 06 2011

All terms start and end with the digit '1'. This fact is used in the given PARI program.

Cf. A020449 - A020472, A062350, A087510 - A087515, A199325 - A199329.

[p: p in PrimesUpTo(10^8) | Set(Intseq(p)) subset [0,1,6] and IsPrime(Seqint(Reverse(Intseq(p))))]; // Bruno Berselli, Nov 07 2011

a(n=50,list=0,L=[0,1,6])={ for(d=1,1e9, my(t,u=vector(d-1,i,10^(d-i))~,o=10^d+1);forvec(v=vector(#u,i,[1,#L]),isprime(t=o+vector(#u,i,L[v[i]])*u) || next; isprime(A004086(t)) || next; list & print1(t", "); n-- || return(t)))}  \\ M. F. Hasler, Nov 07 2011

A036951 Smallest n-digit prime containing only the digits 8 and 9, or 0 if no such prime exists.

Original entry on oeis.org

0, 89, 0, 8999, 89899, 888989, 8888989, 88888999, 888898889, 8888888989, 88888888999, 888888898999, 8888888999899, 88888888888889, 888888888898999, 8888888888989999, 88888888888888889, 888888888888898889

Offset: 1

Patrick De Geest, Jan 04 1999

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

Cf. A020472, A036229, A036325.

Table[SelectFirst[10#+9&/@FromDigits/@Tuples[{8,9},n-1],PrimeQ],{n,20}]/. (Missing["NotFound"]->0) (* Harvey P. Dale, Feb 01 2018 *)

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

A199303 Palindromic primes in the sense of A007500 with digits '0', '1' and '3' only.

Original entry on oeis.org

3, 11, 13, 31, 101, 113, 131, 311, 313, 1031, 1033, 1103, 1301, 3011, 3301, 10301, 10333, 11003, 11311, 13331, 30011, 30103, 31013, 31033, 33013, 33301, 101333, 110311, 113011, 113131, 131311, 133033, 133103, 301331, 301333, 330331, 333101, 333103, 1000033, 1001003, 1001303, 1003001

Offset: 1

M. F. Hasler, Nov 04 2011

Chai Wah Wu, Table of n, a(n) for n = 1..6114

Cf. A020449 - A020472, A199325 - A199329, A199302 - A199306.

[p: p in PrimesUpTo(10^8) | Set(Intseq(p)) subset [0, 1, 3] and IsPrime(Seqint(Reverse(Intseq(p))))]; // Bruno Berselli, Nov 07 2011

Flatten[{#,IntegerReverse[#]}&/@Select[FromDigits/@Tuples[{0,1,3},7],AllTrue[ {#,IntegerReverse[ #]},PrimeQ]&]]//Union (* Harvey P. Dale, Sep 12 2023 *)

allow=Vec("013"); forprime(p=1, default(primelimit), setminus( Set( Vec( Str( p ))), allow)&next; isprime(A004086(p))&print1(p", ")) /* for illustrative purpose only: better use the code below */

a(n=50,list=0,L=[0,1,3],needpal=1)={ 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; needpal & !isprime(A004086(t)) & next; list & print1(t","); n-- || return(t)))}  \\ M. F. Hasler, Nov 06 2011

from itertools import product
from sympy import isprime
A199303_list = [n for n in (int(''.join(s)) for s in product('013',repeat=12)) if isprime(n) and isprime(int(str(n)[::-1]))] # Chai Wah Wu, Dec 17 2015

A199328 Palindromic primes in the sense of A007500 with digits '0', '1' and '8' only.

Original entry on oeis.org

11, 101, 181, 1181, 1811, 18181, 108881, 110881, 118081, 180181, 180811, 181081, 188011, 188801, 1008001, 1088081, 1110881, 1180811, 1181881, 1808801, 1880111, 1880881, 1881811, 1881881, 10001081, 10001801, 10011101, 10080011, 10101181, 10111001, 10111081, 10180801, 10188811, 10808101, 10810001

Offset: 1

M. F. Hasler, Nov 05 2011

Harvey P. Dale, Table of n, a(n) for n = 1..10000 (First 4188 terms from Chai Wah Wu.)

Intersection of A007500 and A061247.

Cf. A020449-A020472, A199325-A199329.

Select[10#+1&/@FromDigits/@Tuples[{0,1,8},7],AllTrue[{#,IntegerReverse[#]},PrimeQ]&] (* Harvey P. Dale, Mar 28 2025 *)

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]),isprime(t=vector(d,i,L[v[i]])*u)||next;isprime(A004086(t))||next;show&print1(t",");n--||return(t)))}

from itertools import product
from sympy import isprime
A199328_list = [n for n in (int(''.join(s)) for s in product('018',repeat=10)) if isprime(n) and isprime(int(str(n)[::-1]))] # Chai Wah Wu, Dec 17 2015

A263431 Near-repdigit primes with only digits 9 and a single 8 in decimal expansion.

Original entry on oeis.org

89, 8999, 98999, 99989, 989999, 9899999, 89999999, 99899999, 99998999, 99999989, 998999999, 98999999999, 99989999999, 999998999999, 999999999899, 999999999989, 99899999999999, 99999899999999, 99999999899999, 999999899999999, 999999999989999, 999999999999989

Offset: 1

Felix Fröhlich, Oct 18 2015

Felix Fröhlich, Table of n, a(n) for n = 1..200

Cf. A020472, A065074, A069661, A105975, A178007.

Select[Flatten[Table[FromDigits/@Permutations[PadRight[{8},n,9]],{n,15}]],PrimeQ] (* Harvey P. Dale, Mar 29 2020 *)

a002283(n) = 10^n-1
a011557(n) = 10^n
num(n, k) = a002283(n)-a011557(k)
terms(n) = i=0; x=1; while(x > 0, y=x-1; while(y >= 0, if(ispseudoprime(num(x, y)), print1(num(x, y), ", "); i++); if(i==n, break({2})); y--); x++)
terms(30) \\ print initial thirty terms

A154523 Numbers k such that the smallest decimal digit of k equals the smallest decimal digit of prime(k).

Original entry on oeis.org

11, 13, 18, 31, 41, 52, 62, 73, 80, 81, 110, 112, 113, 114, 115, 116, 121, 125, 128, 133, 135, 140, 141, 142, 152, 156, 157, 164, 167, 170, 180, 187, 188, 189, 191, 192, 193, 194, 195, 196, 198, 199, 211, 215, 216, 217, 218, 219, 221, 231, 241, 251, 261, 271

Offset: 1

Juri-Stepan Gerasimov, Jan 11 2009

Natural density 1, since almost all numbers and almost all primes (thanks to the prime number theorem) contain the digit 0.

The first terms with smallest digit 1, 2, and 3 are listed in the Data section. The first with smallest digits 4, 5, and 6 are 644, 758, and 6666, respectively. While there are plenty of primes with no decimal digit smaller than 7 (see A106110), including many primes consisting only of the digits 8 and 9 (the 10th of which is prime(77777) = 989999; cf. A020472), it seems to me that finding a term in this sequence whose smallest digit is 7 or 8 should be a very difficult problem. - Jon E. Schoenfield, Feb 11 2019

			11 is a term because prime(11) =  31 (smallest digits: 1);
13 is a term because prime(13) =  41 (smallest digits: 1);
18 is a term because prime(18) =  61 (smallest digits: 1);
31 is a term because prime(31) = 127 (smallest digits: 1);
41 is a term because prime(41) = 179 (smallest digits: 1);
52 is a term because prime(52) = 239 (smallest digits: 2).

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

Cf. A020472, A106110.

A054054 := proc(n) min(op(convert(n,base,10)) ) ; end proc:
for n from 1 to 500 do if A054054(n) = A054054(ithprime(n)) then printf("%d,",n ) ; end if; end do: (End) # R. J. Mathar, May 05 2010

Transpose[Select[Table[{n,Prime[n]},{n,300}],Min[IntegerDigits[#[[1]]]] == Min[IntegerDigits[#[[2]]]]&]][[1]] (* Harvey P. Dale, Dec 18 2012 *)

Corrected (221 inserted) by R. J. Mathar, May 05 2010

Definition clarified by Harvey P. Dale, Dec 18 2012

A174844 Primes that generate three other primes when 2, 6, and 8, respectively, are subtracted from each digit of their decimal representations.

Original entry on oeis.org

9898898899, 889898999999, 889989889889, 898888889989, 989899998889, 999988988989, 988898889999899, 989998888989889, 98888888989989899, 99999998998988999, 888898989989989999, 888998889889898899, 889888889998888999, 889888898999988989, 889988888998998889

Offset: 1

Rick L. Shepherd, Mar 30 2010

Subsequence of A020472. The primes generated from the subtractions are in A020469, A020458, and A020449, respectively. Final digits are necessarily 9 (here), then 7, 3, and 1. Because leading 8's are permitted in the terms here, the primes generated by subtracting 8's may have fewer digits than the others.

			9898898899 is prime and so are 7676676677, 3232232233, and 1010010011, so it is a term. Although 9349, 9349-2222=7127, 9349-6666=2683, and 9349-8888=461 are four primes, 9349 is not a term as subtracting 6 or 8 from the digits 3 and 4 is not possible (no "borrowing" is permitted).

Robert Israel, Table of n, a(n) for n = 1..1188

Cf. A020472, A020469, A020458, A020449.

Res:= NULL: count:= 0:
for d from 2 while count < 100 do
  v:= (10^d-1)/9;
  for m from 1 to d do
    if m mod 3 <> 0 and 2*d+m mod 3 <> 0 then
      for S in combinat:-choose([$1..(d-2)],m-1) do
        q:= 1+add(10^i,i=S);
        if andmap(isprime, [q, 2*v+q, 6*v+q, 8*v+q]) then
           count:= count+1; Res:= Res, 8*v+q;
        fi
      od;
    fi
  od;
od:
sort([Res]); # Robert Israel, Nov 14 2022

okQ[n_]:=Module[{idn=IntegerDigits[n]},And@@PrimeQ[FromDigits/@ {idn-2, idn-6, idn-8}]]; Select[Flatten[Table[Select[FromDigits/@ Tuples[ {8,9},n], PrimeQ],{n,18}]],okQ] (* Harvey P. Dale, Jul 27 2011 *)

{/* Program based on that of M. F. Hasler in A020472. */
for(nd=1, 20, p=vector(nd, i, 10^(nd-i))~; r=(10^nd-1)/9;
forvec(v=vector(nd, i, [8+(i==nd), 9]), q=v*p; isprime(q) &&
isprime(q-2*r ) && isprime(q-6*r ) && isprime(q-8*r ) && print1(q", ")))}

from sympy import isprime
from itertools import count, islice, product
def agen(): # generator of terms
    for d in count(2):
        subs = list(map(int, ["2"*d, "6"*d, "8"*d]))
        for first in product("89", repeat=d-1):
            t = int("".join(first) + "9")
            if isprime(t) and all(isprime(t-s) for s in subs): yield t
print(list(islice(agen(), 15))) # Michael S. Branicky, Nov 15 2022

A199304 Palindromic primes in the sense of A007500 with digits '0', '1' and '4' only.

Original entry on oeis.org

11, 101, 11411, 100411, 101141, 114001, 114041, 140411, 141101, 1004141, 1010411, 1040141, 1041041, 1100441, 1114111, 1140101, 1144441, 1401401, 1410401, 1411141, 1414001, 1440011, 1444411, 1444441, 10010411, 10011101, 10041011, 10044011

Offset: 1

M. F. Hasler, Nov 04 2011

All terms start and end with the digit 1.

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

Cf. A020449 - A020472, A199325 - A199329, A199302 - A199306.

[p: p in PrimesUpTo(10^8) | Set(Intseq(p)) subset [0,1,4] and IsPrime(Seqint(Reverse(Intseq(p))))];  // Bruno Berselli, Nov 07 2011

F:= proc(d) local A0, A4, Res, q, r;
   Res:= NULL;
   q:= (10^(d+1)-1)/9;
   for A0 in combinat:-powerset({$1..d-1}) do
     for A4 in combinat:-powerset({$1..d-1} minus A0) do
       r:= q - add(10^i,i=A0) + 3*add(10^i,i=A4);
       if isprime(r) and isprime(q - add(10^(d-i),i=A0) + 3*add(10^(d-i),i=A4)) then
          Res:= Res, r
       fi
   od od;
   Res
end proc:
sort([seq(F(d),d=1..7)]); # Robert Israel, May 03 2018

allow=Vec("014");forprime(p=1,default(primelimit),setminus( Set( Vec(Str( p ))),allow)&next;isprime(A004086(p))&print1(p",")) /* better use the more efficient code below */

a(n=50,list=0,L=[0,1,4],needpal=1)={ 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; needpal & !isprime(A004086(t)) & next; list & print1(t","); n-- || return(t)))}  \\ M. F. Hasler, Nov 06 2011

cp's OEIS Frontend

A199302 Palindromic primes in the sense of A007500 with digits '0', '1' and '2' only.

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A199306 Palindromic primes in the sense of A007500 with digits '0', '1' and '6' only.

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Magma

PARI

A036951 Smallest n-digit prime containing only the digits 8 and 9, or 0 if no such prime exists.

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Mathematica

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

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A199303 Palindromic primes in the sense of A007500 with digits '0', '1' and '3' only.

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Magma

Mathematica

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A199328 Palindromic primes in the sense of A007500 with digits '0', '1' and '8' only.

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Mathematica

PARI

Python

A263431 Near-repdigit primes with only digits 9 and a single 8 in decimal expansion.

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Mathematica

PARI

A154523 Numbers k such that the smallest decimal digit of k equals the smallest decimal digit of prime(k).

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