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A016115
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Number of prime palindromes with n digits.
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6
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4, 1, 15, 0, 93, 0, 668, 0, 5172, 0, 42042, 0, 353701, 0, 3036643, 0, 27045226, 0, 239093865, 0, 2158090933, 0, 19742800564, 0
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listen;
history;
text;
internal format)
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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 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
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LINKS
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Table of n, a(n) for n=1..24.
K. S. Brown, On General Palindromic Numbers
Patrick De Geest, World!Of Palindromic Primes
Shyam Sunder Gupta, Palindromic Primes up to 10^19.
Shyam Sunder Gupta, Palindromic Primes up to 10^23.
Eric Weisstein's World of Mathematics, Palindromic Prime.
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FORMULA
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a(2n) = 0 for n > 1. - Chai Wah Wu, Nov 21 2021
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MAPLE
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# 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
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MATHEMATICA
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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 *)
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PROG
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(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
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CROSSREFS
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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
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KEYWORD
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nonn,hard,base,more
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AUTHOR
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Robert G. Wilson v
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EXTENSIONS
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Corrected and extended by Patrick De Geest, Jun 15 1998
a(17) = 27045226 was found in collaboration with Martin Eibl (M.EIBL(AT)LINK-R.de), Carlos Rivera, Warut Roonguthai
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
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STATUS
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approved
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