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A348664 Numbers whose binary expansion is not rich. 1
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,1

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.

LINKS

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

FORMULA

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

EXAMPLE

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.

MATHEMATICA

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

PROG

(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

CROSSREFS

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

Sequence in context: A198981 A259330 A090486 * A228320 A346899 A247921

Adjacent sequences:  A348661 A348662 A348663 * A348665 A348666 A348667

KEYWORD

nonn,base

AUTHOR

Rémy Sigrist, Oct 28 2021

STATUS

approved

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Last modified May 23 17:22 EDT 2022. Contains 353978 sequences. (Running on oeis4.)