cp's OEIS Frontend

A348664 Numbers whose binary expansion is not rich.

%I A348664 #15 Oct 29 2021 13:32:15
%S A348664 203,211,300,308,333,357,395,406,407,419,422,423,459,467,556,564,600,
%T A348664 601,604,616,617,628,653,666,667,669,690,709,714,715,723,741,779,787,
%U A348664 790,791,803,811,812,813,814,815,820,835,838,839,844,845,846,847,851,869
%N A348664 Numbers whose binary expansion is not rich.
%C A348664 A word of length k is "rich" if it contains, as contiguous subsequences, exactly k + 1 distinct palindromes (including the empty word).
%C A348664 There are A225681(k)/2 terms with k binary digits.
%H A348664 Rémy Sigrist, <a href="/A348664/b348664.txt">Table of n, a(n) for n = 1..10000</a>
%F A348664 {k: A137397(k) <= A070939(k)}. - _Michael S. Branicky_, Oct 29 2021
%e A348664 For n = 203:
%e A348664 - the binary expansion of 203 is "11001011" and has 8 binary digits,
%e A348664 - we have the following 8 palindromes: "", "0", "1", "00", "11", "010", "101", "1001"
%e A348664 - so 203 is not rich, and belongs to this sequence.
%e A348664 For n = 204:
%e A348664 - the binary expansion of 204 is "11001100" and has 8 binary digits,
%e A348664 - we have the following 9 palindromes: "", "0", "1", "00", "11", "0110", "1001", "001100", "110011"
%e A348664 - so 204 is rich, and does not belong to this sequence.
%t A348664 Select[Range@1000,Length@Select[Union[Subsequences[s=IntegerDigits[#,2]]],PalindromeQ]<=Length@s&] (* _Giorgos Kalogeropoulos_, Oct 29 2021 *)
%o A348664 (PARI) is(n) = { my (b=binary(n), p=select(w->w==Vecrev(w), setbinop((i,j)->b[i..j],[1..#b]))); #b!=#p }
%o A348664 (Python)
%o A348664 def ispal(s): return s == s[::-1]
%o A348664 def ok(n):
%o A348664   s = bin(n)[2:]
%o A348664   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])))
%o A348664 print([k for k in range(870) if ok(k)]) # _Michael S. Branicky_, Oct 29 2021
%Y A348664 Cf. A206926, A216264, A225681, A070939, A137397.
%K A348664 nonn,base
%O A348664 1,1
%A A348664 _Rémy Sigrist_, Oct 28 2021