A348664 Numbers whose binary expansion is not rich.

203, 211, 300, 308, 333, 357, 395, 406, 407, 419, 422, 423, 459, 467, 556, 564, 600, 601, 604, 616, 617, 628, 653, 666, 667, 669, 690, 709, 714, 715, 723, 741, 779, 787, 790, 791, 803, 811, 812, 813, 814, 815, 820, 835, 838, 839, 844, 845, 846, 847, 851, 869

Offset: 1

Author

Rémy Sigrist, Oct 28 2021

Keywords

• nonn
• base

Comments

A word of length k is "rich" if it contains, as contiguous subsequences, exactly k + 1 distinct palindromes (including the empty word).

There are A225681(k)/2 terms with k binary digits.

Examples

			For n = 203:
- the binary expansion of 203 is "11001011" and has 8 binary digits,
- we have the following 8 palindromes: "", "0", "1", "00", "11", "010", "101", "1001"
- so 203 is not rich, and belongs to this sequence.
For n = 204:
- the binary expansion of 204 is "11001100" and has 8 binary digits,
- we have the following 9 palindromes: "", "0", "1", "00", "11", "0110", "1001", "001100", "110011"
- so 204 is rich, and does not belong to this sequence.
		

Links

• Rémy Sigrist, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A206926, A216264, A225681, A070939, A137397.

Programs

Mathematica

Select[Range@1000,Length@Select[Union[Subsequences[s=IntegerDigits[#,2]]],PalindromeQ]<=Length@s&] (* Giorgos Kalogeropoulos, Oct 29 2021 *)

PARI

is(n) = { my (b=binary(n), p=select(w->w==Vecrev(w), setbinop((i,j)->b[i..j],[1..#b]))); #b!=#p }

Python

def ispal(s): return s == s[::-1]
def ok(n):
  s = bin(n)[2:]
  return len(s) >= 1 + len(set(s[i:j] for i in range(len(s)) for j in range(i+1, len(s)+1) if ispal(s[i:j])))
print([k for k in range(870) if ok(k)]) # Michael S. Branicky, Oct 29 2021

Formula

{k: A137397(k) <= A070939(k)}. - Michael S. Branicky, Oct 29 2021