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
References
- James J. Tattersall, Elementary Number Theory in Nine Chapters, Cambridge University Press, 1999, page 113.
Links
- 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.
Crossrefs
Programs
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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
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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 *)
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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 -
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
Formula
a(2n) = 0 for n > 1. - Chai Wah Wu, Nov 21 2021
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
Comments