A373581 Prime numbers whose base-2 representation is a "nested" palindrome.

3, 7, 31, 73, 127, 443, 1453, 5981, 8191, 19609, 21157, 28123, 29671, 83269, 131071, 262657, 287281, 324217, 354997, 367309, 431947, 456571, 462727, 499663, 524287, 1348901, 1488301, 1715851, 1875751, 5548693, 6298627, 7331323, 7355911, 8093551, 8191903

Offset: 1

Author

James S. DeArmon, Jun 10 2024

Keywords

• nonn
• base

Comments

The definition of "nested" palindrome per A344550 is used: both the right and left halves of the base-2 representation of each term are themselves palindromes. "Half" means ceiling(m/2) for a m-bit term. (By contrast, A240601 uses floor(m/2).)

Examples

			Terms 1,2,and 3 are 3,7,31, with respective base-2 valuations of 11, 111, 11111. The fourth term, 73, is the smallest term containing zeros in the base-2 representation: 1001001. Note the middle bit is shared by both halves; the nested palindrome is "1001".
		

Links

• Michael S. Branicky, Table of n, a(n) for n = 1..10000
• James S. DeArmon, Common LISP code for A373581

Crossrefs

Subsequence of A016041.

Primes in A373941.

Cf. A344550.

Programs

Python

import sympy
def ispal(n):
    return str(n) == str(n)[::-1]
def isodd(n): return n%2
outVec = []
for n in range(9999999):
    if not sympy.isprime(n): continue
    binStr = (bin(n))[2:]
    if not ispal(binStr): continue
    lenB = len(binStr)
    halfB = int(lenB/2)
    if isodd(lenB): halfB += 1
    if not ispal(binStr[:halfB]): continue
    print(n,binStr)
    outVec.append(n)
print(outVec)

Python

from sympy import isprime
from itertools import count, islice, product
def pals(d, base=10): # returns a string
    digits = "".join(str(i) for i in range(base))
    for p in product(digits, repeat=d//2):
        if d//2 > 0 and p[0] == "0": continue
        left = "".join(p); right = left[::-1]
        for mid in [[""], digits][d%2]:
            yield left + mid + right
def nbp_gen(): # generator of nested binary palindromes (as strings)
    yield '0'
    for d in count(1):
        yield from (p+p[-1-d%2::-1] for p in pals((d+1)//2, base=2))
def agen(): # generator of terms
    yield from filter(isprime, (int(nbp, 2) for nbp in nbp_gen()))
print(list(islice(agen(), 36))) # Michael S. Branicky, Jun 12 2024
(Common Lisp) ; See Links section.