This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
11011 is in the sequence since it is a palindrome of 5 digits, and the first floor(5/2) digits of it, 11, is also a term. 1001 and 10001 are not in a(n) since 10 is not in a(n).
FromDigits /@ Select[IntegerDigits[Range[2^12], 2], And[PalindromeQ@ Take[#, Floor[Length[#]/2]], PalindromeQ[#]] &] (* Michael De Vlieger, Nov 08 2017 *)
2222 is a nested palindrome since 22=22, and taking just one side, 2=2. For an example using an odd number of digits, 68686 is a nested palindrome since 686 is a palindrome. Using parentheses to indicate nesting: ( (22) (22) ) and ( (686) (686) ), where, in the second example, the middle-most 6 is repeated for exposition.
Select[Range[0,10^5],(k=#;f=FromDigits/@(Take[IntegerDigits[k],#]&/@{l=Ceiling[IntegerLength@k/2],-l});
And@@PalindromeQ/@Join[f,{k}])&] (* Giorgos Kalogeropoulos, Jun 24 2021 *)
Select[Range[0,25000],AllTrue[{#,FromDigits[Take[IntegerDigits[#],Ceiling[ IntegerLength[ #]/2]]]},PalindromeQ]&] (* Harvey P. Dale, May 15 2022 *)
foreach $cand (0..200000){
@a=split("",$cand);
$b = join("",reverse @a);
next unless $cand==$b; # palindromes only
$len = int(@a/2.); $lenA = @a;
$len++ unless ($lenA/2 == int $lenA/2); # an even half? or include middle digit
$half = join("",@a[0..($len-1)]);
$revHalf = join("",reverse @a[0..($len-1)]);
next unless $half == $revHalf;
$str .= "$cand, ";
}
chop $str; chop $str; # remove trailing comma and space
print "$str\n"; # write to stdout
from itertools import 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]:
if left + mid + right != '0': yield left + mid + right
def auptod(dd):
yield 0
for d in range(1, dd+1):
yield from (int(p+p[-1-d%2::-1]) for p in pals((d+1)//2))
print([np for np in auptod(6)]) # Michael S. Branicky, May 22 2021
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".
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.
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