A332661 a(1)=1; a(n+1) is the smallest palindrome not already in the sequence such that the product of a(n+1) and a(n) is also a palindrome.

1, 2, 3, 11, 4, 22, 101, 5, 111, 6, 1001, 7, 88, 77, 8, 1111, 9, 10001, 33, 121, 131, 202, 44, 2002, 141, 212, 1221, 303, 2112, 222, 3003, 232, 10101, 55, 99, 555, 979, 5555, 9779, 55555, 97779, 100001, 66, 1000001, 151, 11011, 161, 11111, 171, 101101, 181

Offset

1,2

Comments

First mentioned by Eric Angelini in the 'math-fun' mailing list on Jan 12, 2020.

The sequence is infinite because, when all smaller numbers fail, a sufficiently large 10^k+1 will always succeed.

Links

Hans Havermann, Table of n, a(n) for n = 1..461

Hans Havermann, Sequence graph using alternately-colored indices

Example

a(4) = 11 because the smallest not-yet-used palindromes (4, 5, 6, 7, 8, 9) multiplied by 3 are not base-10 palindromes, but 11*3 is.

Prog

(Python)
from itertools import count, islice, product
def ispal(n): s = str(n); return s == s[::-1]
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(): # generator of terms
    an, aset = 1, {0, 1} # 0 is not allowed
    yield 1
    while True:
        for p in pals():
            if p not in aset and ispal(an*p):
                an = p; aset.add(an); yield an; break
print(list(islice(agen(), 51))) # Michael S. Branicky, Apr 29 2022

Crossrefs

Cf. A002113 (base-10 palindromes).

Sequence in context: A336876 A083664 A083125 * A031335 A084743 A030391

Adjacent sequences: A332658 A332659 A332660 * A332662 A332663 A332664

Keyword

nonn,base

Author

Eric Angelini and Hans Havermann, Feb 18 2020

Status

approved