A007500 - OEIS

%I M0657 #60 Jan 02 2025 09:33:34

%S 2,3,5,7,11,13,17,31,37,71,73,79,97,101,107,113,131,149,151,157,167,

%T 179,181,191,199,311,313,337,347,353,359,373,383,389,701,709,727,733,

%U 739,743,751,757,761,769,787,797,907,919,929,937,941,953,967,971,983,991,1009,1021

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

%C The numbers themselves need not be palindromes.

%C The range is a subset of the range of A071786. - _Reinhard Zumkeller_, Jul 06 2009

%C 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 Roozbeh Hazrat, Mathematica: A Problem-Centered Approach, Springer 2010, pp. 39, 131-132

%D N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

%D D. Wells, The Penguin Dictionary of Curious and Interesting Numbers. Penguin Books, NY, 1986, 134.

%H T. D. Noe, <a href="/A007500/b007500.txt">Table of n, a(n) for n = 1..10000</a>

%H 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 revdigs:= proc(n)

%p local L,nL,i;

%p L:= convert(n,base,10);

%p nL:= nops(L);

%p add(L[i]*10^(nL-i),i=1..nL);

%p end:

%p Primes:= select(isprime,{2,seq(2*i+1,i=1..5*10^5)}):

%p Primes intersect map(revdigs,Primes); # _Robert Israel_, Aug 14 2014

%t Select[ Prime[ Range[ 168 ] ], PrimeQ[ FromDigits[ Reverse[ IntegerDigits[ # ] ] ] ]& ] (* _Zak Seidov_, corrected by _T. D. Noe_ *)

%t Select[Prime[Range[1000]],PrimeQ[IntegerReverse[#]]&] (* Requires Mathematica version 10 or later *) (* _Harvey P. Dale_, Aug 15 2016 *)

%o (Magma) [ p: p in PrimesUpTo(1030) | IsPrime(Seqint(Reverse(Intseq(p)))) ];  // _Bruno Berselli_, Jul 08 2011

%o (Haskell)

%o a007500 n = a007500_list !! (n-1)

%o a007500_list = filter ((== 1) . a010051 . a004086) a000040_list

%o -- _Reinhard Zumkeller_, Oct 14 2011

%o (PARI) is_A007500(n)={ isprime(n) & is_A095179(n)} \\ _M. F. Hasler_, Jan 13 2012

%o (Python)

%o from sympy import prime, isprime

%o A007500 = [prime(n) for n in range(1,10**6) if isprime(int(str(prime(n))[::-1]))] # _Chai Wah Wu_, Aug 14 2014

%o (Python)

%o from gmpy2 import is_prime, mpz

%o from itertools import count, islice, product

%o def agen(): # generator of terms

%o     yield from [2, 3, 5, 7]

%o     p = 11

%o     for digits in count(2):

%o         for first in "1379":

%o             for mid in product("0123456789", repeat=digits-2):

%o                 for last in "1379":

%o                     s = first + "".join(mid) + last

%o                     if is_prime(t:=mpz(s)) and is_prime(mpz(s[::-1])):

%o                         yield int(t)

%o print(list(islice(agen(), 60))) # _Michael S. Branicky_, Jan 02 2025

%Y Cf. A006567, A007628.

%Y Cf. A002385 (primes that are palindromes in base 10).

%Y Equals A002385 union A006567.

%Y Complement of A076056 with respect to A000040. [From _Reinhard Zumkeller_, Jul 06 2009]

%Y Cf. A004086, A010051, A000040.

%K base,nonn,nice

%O 1,1

%A _N. J. A. Sloane_, _Robert G. Wilson v_

%E More terms from Larry Reeves (larryr(AT)acm.org), Oct 31 2000

%E 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 Third reference added by _Harvey P. Dale_, Oct 17 2011