A356177 Palindromes in A333369.

1, 3, 5, 7, 9, 22, 44, 66, 88, 111, 212, 232, 252, 272, 292, 333, 414, 434, 454, 474, 494, 555, 616, 636, 656, 676, 696, 777, 818, 838, 858, 878, 898, 999, 2002, 2222, 2442, 2662, 2882, 4004, 4224, 4444, 4664, 4884, 6006, 6226, 6446, 6666, 6886, 8008, 8228, 8448, 8668, 8888, 10101

Offset: 1

Author

Bernard Schott, Jul 28 2022

Keywords

• nonn
• base

Comments

If a term has a decimal digit that is odd, it must have an odd number of decimal digits and all odd digits are the same. - Chai Wah Wu, Jul 29 2022

If a term has an even number of decimal digits, then it must have only even decimal digits. - Bernard Schott, Jul 30 2022

Examples

			474 is palindrome and 474 has two 4's and one 7 in its decimal expansion, hence 474 is a term.
		

Links

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

Crossrefs

Intersection of A002113 and A333369.

Cf. A355770, A355771, A355772, A100706 (subsequence of repunits).

Programs

Mathematica

simQ[n_] := AllTrue[Tally @ IntegerDigits[n], EvenQ[Plus @@ #] &]; Select[Range[10^4], PalindromeQ[#] && simQ[#] &] (* Amiram Eldar, Jul 28 2022 *)

Python

from itertools import count, islice, product
def simb(n): s = str(n); return all(s.count(d)%2==int(d)%2 for d in set(s))
def pals(): # generator of palindromes
    digits = "0123456789"
    for d in count(1):
        for p in product(digits, repeat=d//2):
            if d > 1 and p[0] == "0": continue
            left = "".join(p); right = left[::-1]
            for mid in [[""], digits][d%2]:
                yield int(left + mid + right)
def agen(): yield from filter(simb, pals())
print(list(islice(agen(), 55))) # Michael S. Branicky, Jul 28 2022

Python

# faster version based on Comments
from itertools import count, islice, product
def odgen(d): yield from [1, 3, 5, 7, 9] if d == 1 else sorted(int(f+"".join(p)+o+"".join(p[::-1])+f) for o in "13579" for f in o + "2468" for p in product(o+"02468", repeat=d//2-1))
def evgen(d): yield from (int(f+"".join(p)+"".join(p[::-1])+f) for f in "2468" for p in product("02468", repeat=d//2-1))
def A356177gen():
    for d in count(1, step=2): yield from odgen(d); yield from evgen(d+1)
print(list(islice(A356177gen(), 55))) # Michael S. Branicky, Jul 30 2022