A007500 -id:A007500 - cp's OEIS frontend

A111383 Beginning with 3, least member of A007500 such that concatenation of first n terms and its digit reversal both are primes.

3, 7, 3, 3, 79, 701, 157, 1103, 11959, 1901, 10273, 92753, 17047, 11909, 144973, 327251, 99289, 92831, 90373, 309671, 1149619, 745397, 1232083, 94793, 18481, 76607, 186649, 181421, 1657561, 3746111, 7067239, 324143, 3185263, 9457181, 1703413, 3517583, 72481, 12859481

Offset: 1

Hans Havermann, Nov 08 2005

Michael S. Branicky, Table of n, a(n) for n = 1..164

Cf. A113584, A111382, A007500.

from gmpy2 import digits, is_prime, mpz
from itertools import count, islice, product
def bgen(): # generator of terms of A007500 -{2, 5} as strings
    yield from "37"
    p = 11
    for digits in count(2):
        for first in "1379":
            for mid in product("0123456789", repeat=digits-2):
                for last in "1379":
                    s = first + "".join(mid) + last
                    if is_prime(mpz(s)) and is_prime(mpz(s[::-1])):
                        yield s
def agen(): # generator of terms
    s, r, an, san = "", "", 3, "3"
    while True:
        yield int(an)
        s, r = s+san, san[::-1]+r
        for san in bgen():
            if is_prime(mpz(s+san)) and is_prime(mpz(san[::-1]+r)):
                break
        an = mpz(san)
print(list(islice(agen(), 34))) # Michael S. Branicky, Jan 02 2025

a(35) and beyond from Michael S. Branicky, Jan 02 2025

A167517 Emirps (A007500) which are concatenation of three consecutive primes (A030469).

Original entry on oeis.org

353359367, 193319491951, 334733593361, 344934573461, 346734693491, 732173317333, 902990419043, 104591046310477, 133091331313327, 141591417314177, 146571466914683, 150131501715031, 154431545115461

Offset: 1

Jonathan vos Post and M. F. Hasler, Nov 10 2009

A subsequence of A007500, A030469, A132903.

for(i=1,9999, isprime(eval(p=Str(prime(i),prime(i+1),prime(i+2)))) & isprime(eval(concat(vecextract(Vec(p),"-1..1"))))& print1(p,", "))

A167517 = A007500 n A030469 = A007500 n A132903 (where "n" means intersection).

A122490 Least number k>1 such that k+10^n is a symmetric prime with symmetric digits (i.e. such that k+10^n is in A007500).

Original entry on oeis.org

3, 7, 9, 7, 49, 33, 169, 7, 7, 207, 237, 91, 313, 261, 273, 79, 49, 2901, 51, 441, 193, 9, 531, 289, 1141, 67, 909, 331, 753, 2613, 657, 49, 4459, 603, 1531, 849, 2049, 259, 649, 2119, 1483, 63, 6747, 519, 3133, 937, 1159, 1999, 6921, 2949, 613, 4137, 1977, 31, 483, 883, 8553, 12117, 1009, 4347, 733

Offset: 1

Pierre CAMI, Sep 16 2006

			3+10^1=13, 13 and 31 are symmetric primes with symmetric digits;
7+10^2=107, 107 and 701 are symmetric primes with symmetric digits;
9+10^3=1009, 1009 and 9001 are symmetric primes with symmetric digits.

Robert Israel, Table of n, a(n) for n = 1..722

Cf. A007500

revdigs:= proc(n) local L,i;
  L:= convert(n,base,10);
  add(L[-i]*10^(i-1),i=1..nops(L))
end proc:
f:= proc(n) local k,m;
   for k from 3 by 2 do
     m:= k+10^n;
     if isprime(m) and isprime(revdigs(m)) then return k fi
   od
end proc:
map(f, [$1..100]); # Robert Israel, Nov 09 2017

Definition clarified, a(28) corrected, and more terms added by Robert Israel, Nov 09 2017

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

Original entry on oeis.org

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

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

A167518 Least reversible prime (A007500) which is a concatenation of n consecutive primes.

Original entry on oeis.org

2, 151157, 353359367, 139149151157, 101103107109113, 704517045770459704817048770489, 97101103107109113127, 1519315199152171522715233152411525915263, 382138233833384738513853386338773881, 9319932393379341934393499371937793919397

Offset: 1

M. F. Hasler, Nov 10 2009

Here the weaker definition of A007500 is used, but all terms > 2 known so far are also Emirps in the sense of A006567 (i.e. different from their reversal), so it is sufficient to change the first term to 13 in order to have a sequence of "true" emirps.

Is it possible to prove that all terms > 2 are in A006567?

Cf. A030997, A167517.

for(k=1,19,for(i=0,1e9, isprime( eval( p=concat( vector( k,j,Str( prime( i+j )))))) & isprime(eval(concat(vecextract(Vec(p),"-1..1")))) & break); print1(p,", "))

Edited by Charles R Greathouse IV, Apr 28 2010

a(9)-a(10) from Donovan Johnson, Sep 25 2011

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

A135436 a(n) is the least prime for which the n-th term of the sequence S(a(n)) belongs to A007500, where each term of S(p) is the least prime >= the reversal of the previous term.

Original entry on oeis.org

2, 19, 83, 223, 277, 499, 1327, 1747, 2857, 11351, 10831, 11801, 12239, 12211, 18127, 21787, 36709, 30763, 16703

Offset: 1

Philippe LALLOUET (philip.lallouet(AT)orange.fr), Feb 18 2008

After a term of A007500 has appeared in S(p), either this number, if it's truly palindromic, or the pair constituted by it and its reversal, is repeated indefinitely.

For all primes <= 189989, a term of A007500 appears always in S(p) but I could not go further as in the sequence S(p) of the next prime appears a term > 10^6 which is beyond my capacities of calculation. Anyway it's not a surprise and very probably all sequences S(p) reach a stability in a finite limit. What is more surprising is that on the one hand the same term of A007500 appears in sequence S(n) for a(13) and a(14) and on the other hand another same term of A007500 appears in these sequences for a(16), a(17), a(18) and a(19).

			The sequence S(223) is 223, 331, 137, 733 = A007500(38) and that is wrong for any prime lower than 223. Hence a(4)= 223.

Cf. A007500.

cp's OEIS Frontend

A111383 Beginning with 3, least member of A007500 such that concatenation of first n terms and its digit reversal both are primes.

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A167517 Emirps (A007500) which are concatenation of three consecutive primes (A030469).

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A122490 Least number k>1 such that k+10^n is a symmetric prime with symmetric digits (i.e. such that k+10^n is in A007500).

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

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Magma

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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

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

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Magma

Mathematica

PARI

PARI

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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

A167518 Least reversible prime (A007500) which is a concatenation of n consecutive primes.

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PARI

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