A007500 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. (Formerly M0657)

2, 3, 5, 7, 11, 13, 17, 31, 37, 71, 73, 79, 97, 101, 107, 113, 131, 149, 151, 157, 167, 179, 181, 191, 199, 311, 313, 337, 347, 353, 359, 373, 383, 389, 701, 709, 727, 733, 739, 743, 751, 757, 761, 769, 787, 797, 907, 919, 929, 937, 941, 953, 967, 971, 983, 991, 1009, 1021

Offset

1,1

Comments

The numbers themselves need not be palindromes.

The range is a subset of the range of A071786. - Reinhard Zumkeller, Jul 06 2009

Number of terms less than 10^n: 4, 13, 56, 260, 1759, 11297, 82439, 618017, 4815213, 38434593, ..., . - Robert G. Wilson v, Jan 08 2015

References

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

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

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

Links

T. D. Noe, Table of n, a(n) for n = 1..10000

Cécile Dartyge, Bruno Martin, Joël Rivat, Igor E. Shparlinski, and Cathy Swaenepoel, Reversible primes, arXiv:2309.11380 [math.NT], 2023. See p. 3.

Maple

revdigs:= proc(n)
local L, nL, i;
L:= convert(n, base, 10);
nL:= nops(L);
add(L[i]*10^(nL-i), i=1..nL);
end:
Primes:= select(isprime, {2, seq(2*i+1, i=1..5*10^5)}):
Primes intersect map(revdigs, Primes); # Robert Israel, Aug 14 2014

Mathematica

Select[ Prime[ Range[ 168 ] ], PrimeQ[ FromDigits[ Reverse[ IntegerDigits[ # ] ] ] ]& ] (* Zak Seidov, corrected by T. D. Noe *)
Select[Prime[Range[1000]], PrimeQ[IntegerReverse[#]]&] (* Requires Mathematica version 10 or later *) (* Harvey P. Dale, Aug 15 2016 *)

Prog

(Magma) [ p: p in PrimesUpTo(1030) | IsPrime(Seqint(Reverse(Intseq(p)))) ];  // Bruno Berselli, Jul 08 2011
(Haskell)
a007500 n = a007500_list !! (n-1)
a007500_list = filter ((== 1) . a010051 . a004086) a000040_list
-- Reinhard Zumkeller, Oct 14 2011
(PARI) is_A007500(n)={ isprime(n) & is_A095179(n)} \\ M. F. Hasler, Jan 13 2012
(Python)
from sympy import prime, isprime
A007500 = [prime(n) for n in range(1, 10**6) if isprime(int(str(prime(n))[::-1]))] # Chai Wah Wu, Aug 14 2014

Crossrefs

Cf. A006567, A007628.

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

Equals A002385 union A006567.

Complement of A076056 with respect to A000040. [From Reinhard Zumkeller, Jul 06 2009]

Cf. A004086, A010051, A000040.

Sequence in context: A061461 A359139 A069706 * A090933 A107845 A234901

Adjacent sequences: A007497 A007498 A007499 * A007501 A007502 A007503

Keyword

base,nonn,nice

Author

N. J. A. Sloane, Robert G. Wilson v

Extensions

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

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.

Third reference added by Harvey P. Dale, Oct 17 2011

Status

approved