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A002385
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Palindromic primes: prime numbers whose decimal expansion is a palindrome.
(Formerly M0670 N0247)
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273
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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
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OFFSET
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1,1
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COMMENTS
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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
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
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REFERENCES
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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).
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LINKS
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Lubomira Dvorakova, Stanislav Kruml, and David Ryzak, Antipalindromic numbers, arXiv:2008.06864 [math.CO], 2020. Mentions this sequence.
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FORMULA
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MAPLE
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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:
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MATHEMATICA
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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 *)
NestList[NestWhile[NextPrime, #, ! PalindromeQ[#2] &, 2] &, 2, 41] (* Jan Mangaldan, Jul 01 2020 *)
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PROG
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(Haskell)
a002385 n = a002385_list !! (n-1)
a002385_list = filter ((== 1) . a136522) a000040_list
(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)))]
(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
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CROSSREFS
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KEYWORD
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nonn,base,nice,easy
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AUTHOR
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EXTENSIONS
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More terms from Larry Reeves (larryr(AT)acm.org), Oct 25 2000
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STATUS
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approved
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