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A339607 a(1) = 1, a(n) is the least m not already in the sequence that contains the binary expansion of the binary weight of a(n-1) anywhere within its own binary expansion.

%I A339607 #13 Dec 28 2020 17:51:35
%S A339607 1,2,3,4,5,6,8,7,11,12,9,10,13,14,15,16,17,18,19,22,23,20,21,24,25,26,
%T A339607 27,28,29,32,30,33,34,35,31,37,38,39,36,40,41,43,44,45,48,42,46,49,47,
%U A339607 52,50,51,56,53,57,60,64,54,65,55,58,66,59,61,69,62,74,63
%N A339607 a(1) = 1, a(n) is the least m not already in the sequence that contains the binary expansion of the binary weight of a(n-1) anywhere within its own binary expansion.
%C A339607 Conjecture: permutation of the natural numbers.
%C A339607 A permutation of the integers since n appears at or before index 2^n - 1, the first number with binary weight n. - _Michael S. Branicky_, Dec 16 2020
%H A339607 Michael De Vlieger, <a href="/A339607/b339607.txt">Table of n, a(n) for n = 1..16385</a>
%H A339607 Michael De Vlieger, <a href="/A339607/a339607.png">Plot of (n, b(n)) with b(n) = a(n)-n for 1 <= n <= 2^14</a>, highlighting and labeling maxima and local minima in b(n).
%H A339607 Michael De Vlieger, <a href="/A339607/a339607_1.png">Plot of (n, b(n)) with b(n) = a(n)-n for 1 <= n <= 2^14</a>, color function indicating wt(a(n-1)).
%H A339607 Michael De Vlieger, <a href="/A339607/a339607_2.png">Plot of (n, b(n)) with b(n) = a(n)-n for 1 <= n <= 2^14</a>, color function representing degree of consecutive repetition (persistence) of binary weight.
%H A339607 Wikipedia, <a href="https://en.wikipedia.org/wiki/Hamming_weight">Hamming weight</a>
%H A339607 Wolfram Research, <a href="http://functions.wolfram.com/NumberTheoryFunctions/DigitCount/31/01/ShowAll.html">Numbers in Pascal's triangle</a>
%H A339607 <a href="/index/Bi#binary">Index entries for sequences related to binary expansion of n</a>
%e A339607 Let wt(n) = A000120(n)
%e A339607 a(2) = 2 since wt(a(1)) = wt(1) = 1, and we find "1" at the beginning of the binary expansion of the yet unused 2 = "10"_2.
%e A339607 a(3) = 3 since wt(2) = 1, we find "1" as both first and last bit of yet unused 3 = "11"_2.
%e A339607 a(4) = 4 since wt(3) = 2 = "10"_2, we find yet unused 4 = "100"_2 starts with "10"_2.
%e A339607 a(5) = 5 since wt(4) = 1, and yet unused 5 = "101"_2 both starts and ends with 1.
%e A339607 a(6) = 6 since wt(5) = 2 = "10"_2, we find yet unused 6 = "110"_2 ends with "10"_2.
%e A339607 a(7) = 8 since wt(6) = 2 = "10"_2, we find that the least unused m = 7 only contains 1s in binary. The next term m = 8 furnishes "10"_2 at the start of its binary expansion "1000"_2.
%e A339607 a(8) = 7 since wt(8) = 1, we find "1" in three places in the least unused number m = 7 = "111"_2.
%e A339607 a(9) = 11 since wt(7) = 3 = "11"_2, The next unused numbers 9 and 10 are written "1001"_2 and "1010"_2, respectively. Only when we reach m = 11 = "1011"_2 do we find an unused binary number that contains the word "11"_2, etc.
%t A339607 Nest[Append[#, Block[{k = 1, r = IntegerDigits[DigitCount[#[[-1]], 2, 1], 2]}, While[Nand[FreeQ[#, k], SequenceCount[IntegerDigits[k, 2], r] > 0], k++]; k]] & @@ {#, Length@ #} &, {1}, 2^7]
%o A339607 (Python)
%o A339607 def aupto(n):
%o A339607   alst, used = [1], {1}
%o A339607   for i in range(2, n+1):
%o A339607     binprev = bin(alst[-1])[2:]
%o A339607     binwt = binprev.count("1")
%o A339607     targetstr = bin(binwt)[2:]
%o A339607     morebits, extra, ai = 0, 0, binwt
%o A339607     while ai in used:
%o A339607       morebits += 1
%o A339607       found = False
%o A339607       for k in range(2**morebits):
%o A339607         binstrk = bin(k)[2:]
%o A339607         binstrk = "0"*(morebits-len(binstrk)) + binstrk # pad to length
%o A339607         for msbs in range(morebits+1):
%o A339607           trystr = binstrk[:msbs] + targetstr + binstrk[msbs:]
%o A339607           if trystr[0] == "0": continue
%o A339607           trynum = int(trystr, 2)
%o A339607           if trynum not in used:
%o A339607             if not found: ai = trynum; found = True
%o A339607             else: ai = min(ai, trynum)
%o A339607       if found: break
%o A339607     alst.append(ai); used.add(ai)
%o A339607   return alst    # use alst[n-1] for a(n)
%o A339607 print(aupto(68)) # _Michael S. Branicky_, Dec 16 2020
%Y A339607 Cf. A000120, A338209, A339024.
%K A339607 nonn,base,easy
%O A339607 1,2
%A A339607 _Michael De Vlieger_, Dec 16 2020