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 A002385 Palindromic primes: prime numbers whose decimal expansion is a palindrome. (Formerly M0670 N0247) 273
 2, 3, 5, 7, 11, 101, 131, 151, 181, 191, 313, 353, 373, 383, 727, 757, 787, 797, 919, 929, 10301, 10501, 10601, 11311, 11411, 12421, 12721, 12821, 13331, 13831, 13931, 14341, 14741, 15451, 15551, 16061, 16361, 16561, 16661, 17471, 17971, 18181 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,1 COMMENTS 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 This holds in any number base A006093(n), n>1. - Lekraj Beedassy, Mar 07 2005 and Dec 06 2009 The log-log plot shows the fairly regular structure of these numbers. - T. D. Noe, Jul 09 2013 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 Banks, Hart, and Sakata derive a nontrivial upper bound for the number of prime palindromes n <= x as x -> oo. It follows that almost all palindromes are composite. The results hold in any base. The authors use Weil's bound for Kloosterman sums. - Jonathan Sondow, Jan 02 2018 Number of terms < 100^k: 4, 20, 113, 781, 5953, 47995, 401698, ..., . - Robert G. Wilson v, Jan 03 2018 REFERENCES A. H. Beiler, Recreations in the Theory of Numbers, Dover, NY, 1964, p. 228. N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence). N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence). LINKS N. J. A. Sloane, Table of n, a(n) for n = 1..47995 (all palindromic primes with fewer than 12 digits), Oct 14 2015, extending earlier b-files from T. D. Noe and A. Olah. W. D. Banks, D. N. Hart, and M. Sakata, Almost all palindromes are composite, Math. Res. Lett., 11 No. 5-6 (2004), 853-868. K. S. Brown, On General Palindromic Numbers C. K. Caldwell, "Top Twenty" page, Palindrome Patrick De Geest, World!Of Palindromic Primes Lubomira Dvorakova, Stanislav Kruml, and David Ryzak, Antipalindromic numbers, arXiv:2008.06864 [math.CO], 2020. Mentions this sequence. Ernest G. Hibbs, Component Interactions of the Prime Numbers, Ph. D. Thesis, Capitol Technology Univ. (2022), see p. 33. T. D. Noe, Log-log plot of the first 401696 terms I. Peterson, Math Trek, Palindromic Primes Phakhinkon Phunphayap and Prapanpong Pongsriiam, Reciprocal sum of palindromes, arXiv:1803.00161 [math.CA], 2018. M. Shafer, First 401066 Palprimes [Broken link] Eric Weisstein's World of Mathematics, Palindromic Number Eric Weisstein's World of Mathematics, Palindromic Prime Eric Weisstein's World of Mathematics, Integer Sequence Primes Wikipedia, Palindromic prime Index entries for sequences related to palindromes FORMULA Intersection of A000040 (primes) and A002113 (palindromes). A010051(a(n)) * A136522(a(n)) = 1. [Reinhard Zumkeller, Apr 11 2011] Complement of A032350 in A002113. - Jonathan Sondow, Jan 02 2018 MAPLE 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; 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 # A002385 Gets all base-10 palindromic primes with exactly d digits, in the list "Res" d:=7; # (say) if d=1 then Res:= [2, 3, 5, 7]: elif d=2 then Res:= [11]: elif d::even then Res:=[]: else m:= (d-1)/2: Res2 := [seq(seq(n*10^(m+1)+y*10^m+digrev(n), y=0..9), n=10^(m-1)..10^m-1)]: Res:=[]: for x in Res2 do if isprime(x) then Res:=[op(Res), x]; fi: od: fi: Res; # N. J. A. Sloane, Oct 18 2015 MATHEMATICA Select[ Prime[ Range[2100] ], IntegerDigits[#] == Reverse[ IntegerDigits[#] ] & ] 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 *) 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 *) genPal[n_Integer, base_Integer: 10] := Block[{id = IntegerDigits[n, base], insert = Join[{{}}, {# - 1} & /@ Range[base]]}, FromDigits[#, base] & /@ (Join[id, #, Reverse@id] & /@ insert)]; k = 1; lst = {2, 3, 5, 7}; While[k < 19, p = Select[genPal[k], PrimeQ]; If[p != {}, AppendTo[lst, p]]; k++]; Flatten@ lst (* RGWv *) Select[ Prime[ Range[2100]], PalindromeQ] (* Jean-François Alcover, Feb 17 2018 *) NestList[NestWhile[NextPrime, #, ! PalindromeQ[#2] &, 2] &, 2, 41] (* Jan Mangaldan, Jul 01 2020 *) PROG (Haskell) a002385 n = a002385_list !! (n-1) a002385_list = filter ((== 1) . a136522) a000040_list -- Reinhard Zumkeller, Apr 11 2011 (PARI) is(n)=n==eval(concat(Vecrev(Str(n))))&&isprime(n) \\ Charles R Greathouse IV, Nov 20 2012 (PARI) forprime(p=2, 10^5, my(d=digits(p, 10)); if(d==Vecrev(d), print1(p, ", "))); \\ Joerg Arndt, Aug 17 2014 (Python) from itertools import chain from sympy import isprime 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 (Python) from sympy import isprime A002385 = [*filter(isprime, (int(str(x) + str(x)[-2::-1]) for x in range(10**5)))] A002385.insert(4, 11) # Yunhan Shi, Mar 03 2023 (Sage) [n for n in (2..18181) if is_prime(n) and Word(n.digits()).is_palindrome()] # Peter Luschny, Sep 13 2018 (GAP) Filtered([1..20000], n->IsPrime(n) and ListOfDigits(n)=Reversed(ListOfDigits(n))); # Muniru A Asiru, Mar 08 2019 CROSSREFS A007500 = this sequence union A006567. Subsequence of A188650; A188649(a(n)) = a(n); see A033620 for multiplicative closure. [Reinhard Zumkeller, Apr 11 2011] Cf. A016041, A029732, A069469, A117697, A046942, A032350 (Palindromic nonprime numbers). Sequence in context: A083137 A180440 A077652 * A069217 A083139 A088562 Adjacent sequences: A002382 A002383 A002384 * A002386 A002387 A002388 KEYWORD nonn,base,nice,easy AUTHOR N. J. A. Sloane, Simon Plouffe EXTENSIONS More terms from Larry Reeves (larryr(AT)acm.org), Oct 25 2000 Comment from A006093 moved here by Franklin T. Adams-Watters, Dec 03 2009 STATUS approved

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Last modified December 5 13:14 EST 2023. Contains 367591 sequences. (Running on oeis4.)