A344550 -id:A344550 - cp's OEIS frontend

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

James S. DeArmon, Jun 10 2024

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).)

			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".

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

Subsequence of A016041.
Primes in A373941.
Cf. A344550.

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)

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.

A373941 Numbers whose base-2 representation is a "nested" palindrome.

Original entry on oeis.org

0, 1, 3, 7, 15, 21, 31, 45, 63, 73, 127, 153, 255, 273, 341, 443, 511, 561, 693, 891, 1023, 1057, 1453, 1651, 2047, 2145, 2925, 3315, 4095, 4161, 4681, 5461, 5981, 6371, 6891, 7671, 8191, 8385, 9417, 10965, 11997, 12771, 13803, 15351, 16383, 16513, 19609, 21157

Offset: 1

Michael S. Branicky, Jun 23 2024

			45 = 101101_2 is a term, since its base-2 representation is a palindrome, and its right half and left half in base-2 are each the palindrome 101.
73 = 1001001_2 is a term, since its base-2 representation is a palindrome, and its right half and left half (both including the center digit) in base-2 are each the palindrome 1001.

Michael S. Branicky, Table of n, a(n) for n = 1..10000

Base-2 analog of A344550.

Cf. A373581 (prime terms).

isok(k) = my(b=binary(k), b1=Vec(b, ceil(#b/2)), b2=Vec(Vecrev(b), ceil(#b/2))); (Vecrev(b) == b) && (Vecrev(b1) == b1) && (Vecrev(b2) == b2); \\ Michel Marcus, Jun 26 2024

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)
    for d in count(2):
        yield from (p+p[-1-d%2::-1] for p in pals((d+1)//2, base=2))
def agen(): # generator of terms
    yield from [0, 1]
    yield from (int(nbp, 2) for nbp in nbp_gen() if nbp[0] != "0")
print(list(islice(agen(), 46)))

A368020 Palindromes which are a concatenation of three palindromes, each of which has at least 2 digits.

Original entry on oeis.org

110011, 111111, 112211, 113311, 114411, 115511, 116611, 117711, 118811, 119911, 220022, 221122, 222222, 223322, 224422, 225522, 226622, 227722, 228822, 229922, 330033, 331133, 332233, 333333, 334433, 335533, 336633, 337733, 338833, 339933, 440044, 441144, 442244

Offset: 1

James S. DeArmon, Dec 23 2023

Equivalently, these are palindromes which have a palindromic prefix of length at least 2 and no more than 1 less than half the total length. For example, 7 digit terms have the form (aa)(bcb)(aa) and 8 digit terms are of the form (aa)(bccb)(aa) or (aba)(cc)(aba).

			110011 is a term since it is a palindrome, and consists of 3 palindromes: (11)(00)(11).
9999999 is a term and its constituent 3 palindromes can be listed in three ways: (99)(999)(99), (999)(99)(99), and (99)(99)(999).

James S. DeArmon, Python code

Cf. A002113 (palindromes), A344550.

Python
```
# see Link
```

from itertools import count, islice, product
def pals(d=2): # generator of palindromes with d >=2 digits as strings
    yield from (f+(s:="".join(r))+m+s[::-1]+f for f in "123456789" for r in product("0123456789", repeat=d//2-1) for m in [[""], "0123456789"][d%2])
def agen(): # generator of terms
    yield from (int("".join(p)) for d in count(6) for p in pals(d) if any((s:=p[:i])==s[::-1] for i in range(2, d//2)))
print(list(islice(agen(), 33))) # Michael S. Branicky, Jan 23 2024

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

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A373941 Numbers whose base-2 representation is a "nested" palindrome.

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A368020 Palindromes which are a concatenation of three palindromes, each of which has at least 2 digits.

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Author

Keywords

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Examples

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Crossrefs

Programs

Python

Python