cp's OEIS Frontend

A007500 Primes whose reversal in base 10 is also prime (called "palindromic primes" by David Wells, although that name usually refers to A002385). Also called reversible primes.

%I A007500 M0657 #63 Jul 15 2025 11:35:54
%S A007500 2,3,5,7,11,13,17,31,37,71,73,79,97,101,107,113,131,149,151,157,167,
%T A007500 179,181,191,199,311,313,337,347,353,359,373,383,389,701,709,727,733,
%U A007500 739,743,751,757,761,769,787,797,907,919,929,937,941,953,967,971,983,991,1009,1021
%N A007500 Primes whose reversal in base 10 is also prime (called "palindromic primes" by David Wells, although that name usually refers to A002385). Also called reversible primes.
%C A007500 The numbers themselves need not be palindromes.
%C A007500 The range is a subset of the range of A071786. - _Reinhard Zumkeller_, Jul 06 2009
%C A007500 Number of terms less than 10^n: 4, 13, 56, 260, 1759, 11297, 82439, 618017, 4815213, 38434593, ..., . - _Robert G. Wilson v_, Jan 08 2015
%D A007500 Roozbeh Hazrat, Mathematica: A Problem-Centered Approach, Springer 2010, pp. 39, 131-132
%D A007500 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%D A007500 James J. Tattersall, Elementary Number Theory in Nine Chapters, Cambridge University Press, 1999, page 113.
%D A007500 David Wells, The Penguin Dictionary of Curious and Interesting Numbers. Penguin Books, NY, 1986, 134.
%H A007500 T. D. Noe, <a href="/A007500/b007500.txt">Table of n, a(n) for n = 1..10000</a>
%H A007500 Cécile Dartyge, Bruno Martin, Joël Rivat, Igor E. Shparlinski, and Cathy Swaenepoel, <a href="https://arxiv.org/abs/2309.11380">Reversible primes</a>, arXiv:2309.11380 [math.NT], 2023. See p. 3.
%p A007500 revdigs:= proc(n)
%p A007500 local L,nL,i;
%p A007500 L:= convert(n,base,10);
%p A007500 nL:= nops(L);
%p A007500 add(L[i]*10^(nL-i),i=1..nL);
%p A007500 end:
%p A007500 Primes:= select(isprime,{2,seq(2*i+1,i=1..5*10^5)}):
%p A007500 Primes intersect map(revdigs,Primes); # _Robert Israel_, Aug 14 2014
%t A007500 Select[ Prime[ Range[ 168 ] ], PrimeQ[ FromDigits[ Reverse[ IntegerDigits[ # ] ] ] ]& ] (* _Zak Seidov_, corrected by _T. D. Noe_ *)
%t A007500 Select[Prime[Range[1000]],PrimeQ[IntegerReverse[#]]&] (* Requires Mathematica version 10 or later *) (* _Harvey P. Dale_, Aug 15 2016 *)
%o A007500 (Magma) [ p: p in PrimesUpTo(1030) | IsPrime(Seqint(Reverse(Intseq(p)))) ];  // _Bruno Berselli_, Jul 08 2011
%o A007500 (Haskell)
%o A007500 a007500 n = a007500_list !! (n-1)
%o A007500 a007500_list = filter ((== 1) . a010051 . a004086) a000040_list
%o A007500 -- _Reinhard Zumkeller_, Oct 14 2011
%o A007500 (PARI) is_A007500(n)={ isprime(n) & is_A095179(n)} \\ _M. F. Hasler_, Jan 13 2012
%o A007500 (Python)
%o A007500 from sympy import prime, isprime
%o A007500 A007500 = [prime(n) for n in range(1,10**6) if isprime(int(str(prime(n))[::-1]))] # _Chai Wah Wu_, Aug 14 2014
%o A007500 (Python)
%o A007500 from gmpy2 import is_prime, mpz
%o A007500 from itertools import count, islice, product
%o A007500 def agen(): # generator of terms
%o A007500     yield from [2, 3, 5, 7]
%o A007500     p = 11
%o A007500     for digits in count(2):
%o A007500         for first in "1379":
%o A007500             for mid in product("0123456789", repeat=digits-2):
%o A007500                 for last in "1379":
%o A007500                     s = first + "".join(mid) + last
%o A007500                     if is_prime(t:=mpz(s)) and is_prime(mpz(s[::-1])):
%o A007500                         yield int(t)
%o A007500 print(list(islice(agen(), 60))) # _Michael S. Branicky_, Jan 02 2025
%Y A007500 Cf. A006567, A007628.
%Y A007500 Cf. A002385 (primes that are palindromes in base 10).
%Y A007500 Equals A002385 union A006567.
%Y A007500 Complement of A076056 with respect to A000040. [From _Reinhard Zumkeller_, Jul 06 2009]
%Y A007500 Cf. A004086, A010051, A000040.
%K A007500 base,nonn,nice
%O A007500 1,1
%A A007500 _N. J. A. Sloane_, _Robert G. Wilson v_
%E A007500 More terms from Larry Reeves (larryr(AT)acm.org), Oct 31 2000
%E A007500 Added further terms to the sequence Avik Roy (avik_3.1416(AT)yahoo.co.in), Jan 16 2009. Checked by _N. J. A. Sloane_, Jan 20 2009.
%E A007500 Third reference added by _Harvey P. Dale_, Oct 17 2011