A144752 Positive integers whose binary representation is a palindrome and has a prime number of 0's.

9, 17, 21, 45, 51, 65, 85, 93, 99, 107, 189, 219, 231, 257, 297, 325, 365, 381, 387, 427, 443, 455, 471, 765, 891, 951, 975, 1105, 1161, 1241, 1285, 1365, 1421, 1501, 1533, 1539, 1619, 1675, 1755, 1787, 1799, 1879, 1911, 1935, 1967, 3069, 3579, 3831, 3951

Offset: 1

Author

Leroy Quet, Sep 20 2008

Keywords

• base
• nonn

Comments

Each term of this sequence is in both sequence A006995 and sequence A144754.

Examples

			17 in binary is 10001. This binary representation is a palindrome, it contains three 0's, and three is a prime. So 17 is a term.
		

Links

• Indranil Ghosh, Table of n, a(n) for n = 1..11167

Crossrefs

Cf. A144753, A006995, A144754

Programs

Python

from sympy import isprime
def ok(n): b = bin(n)[2:]; return b == b[::-1] and isprime(b.count('0'))
print(list(filter(ok, range(4000)))) # Michael S. Branicky, Sep 17 2021

Python

# faster for computing initial segment of sequence
from sympy import isprime
from itertools import product
def ok2(bin_str): return isprime(bin_str.count("0"))
def bin_pals(maxdigits):
    yield from "01"
    digits, midrange = 2, [[""], ["0", "1"]]
    for digits in range(2, maxdigits+1):
        for p in product("01", repeat=digits//2-1):
            left = "1"+"".join(p)
            for middle in midrange[digits%2]:
                yield left + middle + left[::-1]
def auptopow2(e): return [int(b, 2) for b in filter(ok2, bin_pals(e))]
print(auptopow2(12)) # Michael S. Branicky, Sep 17 2021

Extensions

Extended by Ray Chandler, Nov 04 2008

Name edited by Michael S. Branicky, Sep 17 2021