A016115 Number of prime palindromes with n digits.

4, 1, 15, 0, 93, 0, 668, 0, 5172, 0, 42042, 0, 353701, 0, 3036643, 0, 27045226, 0, 239093865, 0, 2158090933, 0, 19742800564, 0, 180815391365, 0

Offset

1,1

Comments

Every palindrome with an even number of digits is divisible by 11 and therefore is composite (not prime). Hence there is only one palindromic prime with an even number of digits, namely 11 itself. - Martin Renner, Apr 15 2006

Links

Table of n, a(n) for n=1..26.

K. S. Brown, On General Palindromic Numbers

Cécile Dartyge, Bruno Martin, Joël Rivat, Igor E. Shparlinski, and Cathy Swaenepoel, Reversible primes, arXiv:2309.11380 [math.NT], 2023. See p. 36.

Patrick De Geest, World!Of Palindromic Primes

Shyam Sunder Gupta, Palindromic Primes up to 10^19, Feb. 6, 2006.

Shyam Sunder Gupta, Palindromic Primes up to 10^21, March 13, 2009.

Shyam Sunder Gupta, Palindromic Primes up to 10^23, Oct. 4, 2013.

Shyam Sunder Gupta, Palindromic Primes up to 10^25, Dec. 18, 2024.

Eric Weisstein's World of Mathematics, Palindromic Prime.

Formula

a(2n) = 0 for n > 1. - Chai Wah Wu, Nov 21 2021

Maple

# A016115 Gets numbers of base-10 palindromic primes with exactly d digits, 1 <= d <= 13 (say), in the list "lis"
lis:=[4, 1];
for d from 3 to 13 do
if d::even then
    lis:=[op(lis), 0];
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)]:
    ct:=0; for x in Res2 do if isprime(x) then ct:=ct+1; fi: od:
    lis:=[op(lis), ct];
fi:
lprint(d, lis);
od:
lis; # N. J. A. Sloane, Oct 18 2015

Mathematica

A016115[n_] := Module[{i}, If[EvenQ[n] && n > 2, Return[0]]; Return[Length[Select[Range[10^(n - 1), 10^n - 1], # == IntegerReverse[#] && PrimeQ[#] &]]]];
Table[A016115[n], {n, 6}] (* Robert Price, May 25 2019 *)
(* -OR-  A less straightforward implementation, but more efficient in that the palindromes are constructed instead of testing every number in the range. *)
A016115[n_] := Module[{c, f, t0, t1},
   If[n == 2, Return[1]];
   If[EvenQ[n], Return[0]];
   c = 0; t0 = 10^((n - 1)/2); t1 = t0*10;
   For[f = t0, f < t1, f++,
    If[n != 1 && MemberQ[{2, 4, 5, 6, 8}, Floor[f/t0]], f = f + t0 - 1; Continue[]];
    If[PrimeQ[f*t0 + IntegerReverse[Floor[f/10]]], c++]]; Return[c]];
Table[A016115[n], {n, 1, 12}] (* Robert Price, May 25 2019 *)

Prog

(Python)
from sympy import isprime
from itertools import product
def pals(d, base=10): # all d-digit palindromes
    digits = "".join(str(i) for i in range(base))
    for p in product(digits, repeat=d//2):
        if d > 1 and p[0] == "0": continue
        left = "".join(p); right = left[::-1]
        for mid in [[""], digits][d%2]: yield int(left + mid + right)
def a(n): return int(n==2) if n%2 == 0 else sum(isprime(p) for p in pals(n))
print([a(n) for n in range(1, 13)]) # Michael S. Branicky, Jun 23 2021
(PARI) apply( {A016115(n)=if(n%2, (n<3)+vecsum([sum(k=i, i+n, (k*2-k%10)%3 && isprime(k*n+fromdigits(Vecrev(digits(k\10))))) | i<-[1, 3, 7, 9]*n=10^(n\2)]), n==2)}, [1..12]) \\ M. F. Hasler, Dec 19 2024

Crossrefs

Cf. A002113 (palindromes), A002385 (palindromic primes), A040025 (bisection), A050251 (partial sums).

Sequence in context: A257501 A096644 A145829 * A164794 A353763 A226478

Adjacent sequences: A016112 A016113 A016114 * A016116 A016117 A016118

Keyword

nonn,hard,base,more

Author

Robert G. Wilson v

Extensions

Corrected and extended by Patrick De Geest, Jun 15 1998

a(17) = 27045226 was found by Martin Eibl (M.EIBL(AT)LINK-R.de) and independently by Warut Roonguthai and later confirmed by Carlos Rivera, in June 1998.

a(19) from Shyam Sunder Gupta, Feb 12 2006

a(21)-a(22) from Shyam Sunder Gupta, Mar 13 2009

a(23)-a(24) from Shyam Sunder Gupta, Oct 05 2013

a(25)-a(26) from Shyam Sunder Gupta, Dec 19 2024

Status

approved