A380596 Numbers with embedded palindromes as proper substrings of the term.

100, 110, 111, 112, 113, 114, 115, 116, 117, 118, 119, 122, 133, 144, 155, 166, 177, 188, 199, 200, 211, 220, 221, 222, 223, 224, 225, 226, 227, 228, 229, 233, 244, 255, 266, 277, 288, 299, 300, 311, 322, 330, 331, 332, 333, 334, 335, 336, 337, 338, 339, 344, 355, 366, 377, 388, 399

Offset

1,1

Comments

An embedded palindrome is a substring of at least two contiguous digits (since a single digit is trivially a palindrome). E.g., 121 is a palindrome, but has no embedded palindromes; 110 has the embedded palindrome "11".

Alternatively, k contains a proper substring of the form dd or ded, where d and e are single decimal digits (i.e., a length-2 or -3 palindrome). - Michael S. Branicky, Feb 08 2025

Links

Table of n, a(n) for n=1..57.

Example

100 is a term, since "00" is a palindrome; 1001 is a term for the same reason.
1020 is a term, since "020" is a palindrome; 10201 is a term for the same reason.

Prog

(Python)
from itertools import combinations
def get_all_substrings(string):
    length = len(string) + 1
    return [string[x:y] for x, y in combinations(range(length), r=2)]
def is_palindrome(n):
    return str(n) == str(n)[::-1]
def ok(n):
    subsets = get_all_substrings(str(n))
    subsets = [subset for subset in subsets if is_palindrome(subset) and len(subset)>1 and len(subset)<len(str(n))]
    return len(subsets)>0
print([k for k in range (100, 400) if ok(k)])
(Python)
def ok(n):
    s = str(n)
    return any(p == p[::-1] and len(p) < len(s) for p in (s[i:i+j] for j in (2, 3) for i in range(len(s)-j+1)))
print([k for k in range(400) if ok(k)]) # Michael S. Branicky, Feb 08 2025

Crossrefs

Cf. A002113.

Significant overlap with A044821 for terms below 1000.

Sequence in context: A352440 A204589 A134496 * A191752 A165406 A135603

Adjacent sequences: A380593 A380594 A380595 * A380597 A380598 A380599

Keyword

nonn,base

Author

James S. DeArmon, Jan 27 2025

Status

approved