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A006567
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Emirps (primes whose reversal is a different prime).
(Formerly M4887)
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196
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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
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OFFSET
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1,1
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COMMENTS
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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
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REFERENCES
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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).
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LINKS
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Chris K. Caldwell, The Prime Glossary, emirp.
Eric Weisstein's World of Mathematics, Emirp.
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MAPLE
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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:
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MATHEMATICA
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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 *)
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PROG
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(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
(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
(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
(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)
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CROSSREFS
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KEYWORD
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nonn,nice,easy,base
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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