A117773 Number of palindromic primes in base 2 with exactly n binary digits.

0, 1, 2, 0, 2, 0, 3, 0, 3, 0, 7, 0, 12, 0, 23, 0, 40, 0, 94, 0, 142, 0, 271, 0, 480, 0, 856, 0, 1721, 0, 3099, 0, 5572, 0, 10799, 0, 20782, 0, 39468, 0, 72672, 0, 139867, 0, 274480, 0, 520376, 0, 986318, 0, 1914097, 0, 3726617, 0, 7107443, 0, 13682325, 0, 26430797, 0, 51412565, 0, 99204128, 0, 190457946, 0

Offset: 1

Author

Martin Renner, Apr 15 2006

Keywords

• nonn
• base

Comments

Every palindrome with an even number of digits is divisible by 11 (in base 2), i.e., by 3 in base 10, and therefore is composite (not prime). Hence there is only one palindromic prime with an even number of digits, namely 11_2 = 3_{10}.

Links

• Chai Wah Wu, Table of n, a(n) for n = 1..76
• Cécile Dartyge, Bruno Martin, Joël Rivat, Igor E. Shparlinski, and Cathy Swaenepoel, Reversible primes, arXiv:2309.11380 [math.NT], 2023. See p. 34.
• Eric Weisstein's World of Mathematics, Palindromic Prime.

Crossrefs

Cf. A016041, A117697, A095741.

Programs

Mathematica

Array[If[And[OddQ[#], # > 1], 0, Count[Prime@ Range[PrimePi[2^#] - Boole[# == 1] + 1, PrimePi[2^(# + 1) - 1]], ?(PalindromeQ[IntegerDigits[#, 2]] &)]] &, 25, 0] (* _Michael De Vlieger, Sep 29 2023 *)

Extensions

a(23)-a(40) from Donovan Johnson, Dec 02 2009

a(41)-a(66) from Martin Raab, Oct 20 2015