A006567 Emirps (primes whose reversal is a different prime).

13, 17, 31, 37, 71, 73, 79, 97, 107, 113, 149, 157, 167, 179, 199, 311, 337, 347, 359, 389, 701, 709, 733, 739, 743, 751, 761, 769, 907, 937, 941, 953, 967, 971, 983, 991, 1009, 1021, 1031, 1033, 1061, 1069, 1091, 1097, 1103, 1109, 1151, 1153, 1181, 1193, 1201

Offset: 1

Author

N. J. A. Sloane

Keywords

• nonn
• nice
• easy
• base

Comments

A palindrome is a word that when written in reverse results in the same word. for example, "racecar" reversed is still "racecar". Related to palindromes are semordnilaps. These are words that when written in reverse result in a distinct valid word. For example, "stressed" written in reverse is "desserts". Not all words are palindromes or semordnilaps. While certainly not all numbers are palindromes, all non-palindromic numbers when written in reverse will form semordnilaps. Narrowing to primes brings back the same trichotomy as with words: some numbers are emirps, some numbers are palindromic primes, but some words are neither.

The term "emirp" was coined by the American mathematician Jeremiah Farrell (1937-2022). - Amiram Eldar, Jun 11 2021

References

• Martin Gardner, The Magic Numbers of Dr Matrix. Prometheus, Buffalo, NY, 1985, p. 230.
• N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Links

• T. D. Noe, Table of n, a(n) for n = 1..10000
• Chris K. Caldwell, The Prime Glossary, emirp.
• Brady Haran and N. J. A. Sloane, What Number Comes Next?, Numberphile video, 2018.
• Eric Weisstein's World of Mathematics, Emirp.
• Wikipedia, Emirp.

Crossrefs

Cf. A003684, A007628 (subsequence), A046732, A048051, A048052, A048053, A048054, A048895, A004086 (read n backwards).

A007500 is the union of A002385 and this sequence.

Programs

Haskell

a006567 n = a006567_list !! (n-1)
a006567_list = filter f a000040_list where
   f p = a010051' q == 1 && q /= p  where q = a004086 p
-- Reinhard Zumkeller, Jul 16 2014

Magma

[ n : n in [1..1194] | n ne rev and IsPrime(n) and IsPrime(rev) where rev is Seqint(Reverse(Intseq(n))) ]; // Sergei Haller (sergei(AT)sergei-haller.de), Dec 21 2006

Maple

read("transforms") ; isA006567 := proc(n) local R ; if isprime(n) then R := digrev(n) ; isprime(R) and R <> n ; else false; end if; end proc:
A006567 := proc(n) option remember ; local a; if n = 1 then 13; else a := nextprime(procname(n-1)) ; while not isA006567(a) do a := nextprime(a) ; end do; return a; end if; end proc:
seq(A006567(n),n=1..120) ; # R. J. Mathar, May 24 2010

Mathematica

fQ[n_] := Block[{idn = IntegerReverse@ n}, PrimeQ@ idn && n != idn]; Select[Prime@ Range@ 200, fQ] (* Santi Spadaro, Oct 14 2001 and modified by Robert G. Wilson v, Nov 08 2015 *)
Select[Prime[Range[5,200]],PrimeQ[IntegerReverse[#]]&&!PalindromeQ[#]&] (* Requires Mathematica version 10 or later *) (* Harvey P. Dale, May 11 2021 *)

PARI

is(n)=my(r=eval(concat(Vecrev(Str(n)))));isprime(r)&&r!=n&&isprime(n) \\ Charles R Greathouse IV, Nov 20 2012

PARI

select( {is_A006567(n,r=fromdigits(Vecrev(digits(n))))=isprime(r)&&r!=n&&isprime(n)}, primes(200)) \\ M. F. Hasler, Jan 31 2020

Python

from sympy import prime, isprime
A006567 = [p for p in (prime(n) for n in range(1,10**6)) if str(p) != str(p)[::-1] and isprime(int(str(p)[::-1]))] # Chai Wah Wu, Aug 14 2014

Python

from sympy import isprime, nextprime
def emirps(start=1, end=float('inf')): # generator for emirps in start..end
    p = nextprime(start-1)
    while p <= end:
        s = str(p)
        if s[0] in "24568":
            p = nextprime((int(s[0])+1)*10**(len(s)-1)); continue
        revp = int(s[::-1])
        if p != revp and isprime(revp): yield p
        p = nextprime(p)
print(list(emirps(end=1201))) # Michael S. Branicky, Jan 24 2021, updated Jul 28 2022

Extensions

More terms from James Sellers, Jan 22 2000