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

1,3

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.

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 *)

Crossrefs

Cf. A016041, A117697, A095741.

Sequence in context: A242096 A284965 A274101 * A363824 A025805 A029192

Adjacent sequences: A117770 A117771 A117772 * A117774 A117775 A117776

Keyword

nonn,base

Author

Martin Renner, Apr 15 2006

Extensions

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

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

Status

approved