This site is supported by donations to The OEIS Foundation.



Annual Appeal: Please make a donation (tax deductible in USA) to keep the OEIS running. Over 5000 articles have referenced us, often saying "we discovered this result with the help of the OEIS".

(Greetings from The On-Line Encyclopedia of Integer Sequences!)
A002385 Palindromic primes: prime numbers whose decimal expansion is a palindrome.
(Formerly M0670 N0247)

%I M0670 N0247

%S 2,3,5,7,11,101,131,151,181,191,313,353,373,383,727,757,787,797,919,

%T 929,10301,10501,10601,11311,11411,12421,12721,12821,13331,13831,

%U 13931,14341,14741,15451,15551,16061,16361,16561,16661,17471,17971,18181

%N Palindromic primes: prime numbers whose decimal expansion is a palindrome.

%C Every palindrome with an even number of digits is divisible by 11, so 11 is the only member of the sequence with an even number of digits. - _David Wasserman_, Sep 09 2004

%C This holds in any number base A006093(n), n>1. - _Lekraj Beedassy_, Mar 07 2005

%C The log-log plot shows the fairly regular structure of these numbers. - _T. D. Noe_, Jul 09 2013

%C Conjecture: The only primes with palindromic prime indices that are palindromic primes themselves are 3, 5 and 11. Tested for the primes with the first 8000000 palindromic prime indices. - _Ivan N. Ianakiev_, Oct 10 2014

%D A. H. Beiler, Recreations in the Theory of Numbers, Dover, NY, 1964, p. 228.

%D N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).

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

%H N. J. A. Sloane, <a href="/A002385/b002385.txt">Table of n, a(n) for n = 1..47995 (all primes with fewer than 12 digits)</a>, Oct 14 2015, extending earlier b-files from T. D. Noe and A. Olah.

%H K. S. Brown, <a href="http://www.mathpages.com/home/kmath359.htm">On General Palindromic Numbers</a>

%H C. K. Caldwell, "Top Twenty" page, <a href="http://primes.utm.edu/top20/page.php?id=53">Palindrome</a>

%H P. De Geest, <a href="http://www.worldofnumbers.com/palpri.htm">World!Of Palindromic Primes</a>

%H T. D. Noe, <a href="/A002385/a002385.jpg">Log-log plot of the first 401696 terms</a>

%H I. Peterson, Math Trek, <a href="http://web.archive.org/web/20130103135005/http://www.maa.org/mathland/mathtrek_5_10_99.html">Palindromic Primes</a>

%H M. Shafer, <a href="http://www.egr.msu.edu/~shafermi/primes">First 401066 Palprimes</a> [Broken link]

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/PalindromicNumber.html">Palindromic Number</a>

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/PalindromicPrime.html">Palindromic Prime</a>

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/IntegerSequencePrimes.html">Integer Sequence Primes</a>

%H Wikipedia, <a href="http://en.wikipedia.org/wiki/Palindromic_prime">Palindromic prime</a>

%H <a href="/index/Pac#palindromes">Index entries for sequences related to palindromes</a>

%F Intersection of A000040 (primes) and A002113 (palindromes).

%F A010051(a(n)) * A136522(a(n)) = 1. [_Reinhard Zumkeller_, Apr 11 2011]

%p ff := proc(n) local i,j,k,s,aa,nn,bb,flag; s := n; aa := convert(s,string); nn := length(aa); bb := ``; for i from nn by -1 to 1 do bb := cat(bb,substring(aa,i..i)); od; flag := 0; for j from 1 to nn do if substring(aa,j..j)<>substring(bb,j..j) then flag := 1 fi; od; RETURN(flag); end; gg := proc(i) if ff(ithprime(i)) = 0 then RETURN(ithprime(i)) fi end;

%p rev:=proc(n) local nn, nnn: nn:=convert(n,base,10): add(nn[nops(nn)+1-j]*10^(j-1),j=1..nops(nn)) end: a:=proc(n) if n=rev(n) and isprime(n)=true then n else fi end: seq(a(n),n=1..20000); # rev is a Maple program to revert a number - _Emeric Deutsch_, Mar 25 2007

%p # A002385 Gets all base-10 palindromic primes with exactly d digits, in the list "Res"

%p d:=7; # (say)

%p if d=1 then Res:= [2,3,5,7]:

%p elif d=2 then Res:= [11]:

%p elif d::even then

%p Res:=[]:

%p else

%p m:= (d-1)/2:

%p Res2 := [seq(seq(n*10^(m+1)+y*10^m+digrev(n), y=0..9), n=10^(m-1)..10^m-1)]:

%p Res:=[]: for x in Res2 do if isprime(x) then Res:=[op(Res),x]; fi: od:

%p fi:

%p Res; # _N. J. A. Sloane_, Oct 18 2015

%t Select[ Prime[ Range[2100] ], IntegerDigits[#] == Reverse[ IntegerDigits[#] ] & ]

%t lst = {}; e = 3; Do[p = n*10^(IntegerLength[n] - 1) + FromDigits@Rest@Reverse@IntegerDigits[n]; If[PrimeQ[p], AppendTo[lst, p]], {n, 10^e - 1}]; Insert[lst, 11, 5] (* _Arkadiusz Wesolowski_, May 04 2012 *)

%t Join[{2,3,5,7,11},Flatten[Table[Select[Prime[Range[PrimePi[ 10^(2n)]+1, PrimePi[ 10^(2n+1)]]],# == IntegerReverse[#]&],{n,3}]]] (* The program uses the IntegerReverse function from Mathematica version 10 *) (* _Harvey P. Dale_, Apr 22 2016 *)

%o (Haskell)

%o a002385 n = a002385_list !! (n-1)

%o a002385_list = filter ((== 1) . a136522) a000040_list

%o -- _Reinhard Zumkeller_, Apr 11 2011

%o (PARI) is(n)=n==eval(concat(Vecrev(Str(n))))&&isprime(n) \\ _Charles R Greathouse IV_, Nov 20 2012

%o (PARI) forprime(p=2,10^5, my(d=digits(p,10)); if(d==Vecrev(d),print1(p,", "))); \\ _Joerg Arndt_, Aug 17 2014

%o (Python)

%o from itertools import chain

%o from sympy import isprime

%o A002385 = sorted((n for n in chain((int(str(x)+str(x)[::-1]) for x in range(1,10**5)),(int(str(x)+str(x)[-2::-1]) for x in range(1,10**5))) if isprime(n))) # _Chai Wah Wu_, Aug 16 2014

%Y A007500 = this sequence union A006567.

%Y Subsequence of A188650; A188649(a(n)) = a(n); see A033620 for multiplicative closure. [_Reinhard Zumkeller_, Apr 11 2011]

%Y Cf. A016041, A029732, A117697.

%K nonn,base,nice,easy

%O 1,1

%A _N. J. A. Sloane_, _Simon Plouffe_

%E More terms from Larry Reeves (larryr(AT)acm.org), Oct 25 2000

%E Comment from A006093 moved here by _Franklin T. Adams-Watters_, Dec 03 2009

%E Mentioned the sequence A006093 in my comment, previously omitted by mistake _Lekraj Beedassy_, Dec 06 2009

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recent | More pages
The OEIS Community | Maintained by The OEIS Foundation Inc.

License Agreements, Terms of Use, Privacy Policy .

Last modified December 5 05:27 EST 2016. Contains 278761 sequences.