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

Author

James S. DeArmon, Jan 27 2025

Keywords

• nonn
• base

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

Examples

			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.
		

Crossrefs

Cf. A002113.

Significant overlap with A044821 for terms below 1000.

Programs

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)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