A117697 Palindromic primes in base 2 (written in base 2).

11, 101, 111, 10001, 11111, 1001001, 1101011, 1111111, 100000001, 100111001, 110111011, 10010101001, 10110101101, 11000100011, 11001010011, 11011111011, 11100100111, 11101010111, 1001100011001, 1001111111001, 1010001000101, 1010110110101, 1011011101101, 1011101011101

Offset

1,1

Links

Attila Olah, Table of n, a(n) for n=1..43964

Lubomira Dvorakova, Stanislav Kruml, and David Ryzak, Antipalindromic numbers, arXiv:2008.06864 [math.CO], 2020. Mentions this sequence.

R. Potůček, On the Relation Between Binary Palindromes with Three Alternate Blocks and Fermat Numbers, Equations 5 (2025), 100-105. See p. 100.

Eric Weisstein's World of Mathematics, Palindromic Prime.

Formula

a(n) = A007088(A016041(n)). - R. J. Mathar, Feb 25 2007

Maple

isA016041 := proc(n)
    local bin, dig ;
    if isprime(n) then
        bin := convert(n, base, 2) ;
        for dig from 1 to nops(bin)/2 do
            if op(dig, bin) <> op(-dig, bin) then
                return false;
            end if;
        end do ;
        return true;
    else
        false ;
    end if ;
end proc:
for i from 1 to 900 do p := ithprime(i) : if isA016041(p) then printf("%d, ", A007088(p)) ; fi ; od : # R. J. Mathar, Feb 25 2007

Mathematica

pal2Q[n_] := Reverse[x = IntegerDigits[n, 2]] == x; BaseForm[Select[Prime[Range[700]], pal2Q[#] &], 2] (* Jayanta Basu, Jun 24 2013 *)
(* Alternative: *)
FromDigits /@ Select[IntegerDigits[Prime@ Range[1000], 2], PalindromeQ] (* Michael De Vlieger, Oct 28 2020 *)

Prog

(Python)
from sympy import isprime
from itertools import count, islice, product
def agen(): # generator of terms
    yield 11
    for d in count(3, 2):
        for rest in product("01", repeat=d//2-1):
            left = "1" + "".join(rest)
            for mid in "01":
                if isprime(int(left + mid + left[::-1], 2)):
                    yield int(left + mid + left[::-1])
print(list(islice(agen(), 24))) # Michael S. Branicky, Jun 23 2026

Crossrefs

Cf. A016041, A002385.

Sequence in context: A350346 A247647 A240602 * A305395 A158051 A091366

Adjacent sequences: A117694 A117695 A117696 * A117698 A117699 A117700

Keyword

nonn,base

Author

Martin Renner, Apr 13 2006

Status

approved