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 A339024 a(1) = 1, a(n) is the least m not already in the sequence whose binary expansion begins with the binary expansion of the binary weight of a(n-1). 3
 1, 2, 3, 4, 5, 8, 6, 9, 10, 11, 7, 12, 16, 13, 14, 15, 17, 18, 19, 24, 20, 21, 25, 26, 27, 32, 22, 28, 29, 33, 23, 34, 35, 30, 36, 37, 31, 40, 38, 48, 39, 64, 41, 49, 50, 51, 65, 42, 52, 53, 66, 43, 67, 54, 68, 44, 55, 45, 69, 56, 57, 70, 58, 71, 72, 46, 73, 59 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,2 COMMENTS We define binary weight wt(n) = A000120(n) as the number of 1s in n_2, the number n expressed in binary. Let w = wt(a(n-1)) the binary weight of the previous term, where w_2 is w expressed in binary, and let interval I(j) = 2^j <= n <= (2^(j+1)-1).' Likely a permutation of the natural numbers. The plot (n, a(n)) is organized into streaky clouds that pertains to a "family" M(i) <= m < M(i+1) whose binary expansion begins with an odd "prefix" m/2^v, where v is the 2-adic valuation of m. There are thus 2^v numbers in this range. The numbers in this range accommodate the binary weights wt(a(n-1)) = w with 1 <= w <= ceiling(log_2 a(n-1)) such that w_2 appears in part or all of the binary expansion of the prefix m/2^v, and perhaps an additional bit in m after the prefix. Small values of w, for instance w = 1, may appear in any family, but large w require the entire prefix and potentially more (if even). The w that cannot be found in a particular family are found in a different family that has M(i+1) as its least member. The families M(i) belong in turn to classes according to odd prefixes. Thus, for example, we may find w = 1, 2, 4, and 9 in class 9, since "1", "10", "100", and "1001" can be found in numbers m that begin, "1001...". For w in interval I(j), we have values 1 <= k <= j - 1 distributed binomially. Permutation of the natural numbers. We can always find w in a number m in family M(i) that pertains to a class C of numbers that in binary start with the binary expansion of an odd number c. Numbers m that begin with numbers that are formed of left-trimmed bits of c exhaust the numbers in M(i) before moving to M(i+1) in the same class C. When we have recordsetting odd w, a new class C opens up based on the binary expansion of a larger odd number c. 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 LINKS Michael De Vlieger, Table of n, a(n) for n = 1..16384 Michael De Vlieger, Plot (n, a(n)) for 1 <= n <= 2^10 color-coded to show wt(a(n-1)), with the first term in the family indicated. Michael De Vlieger, Plot (n, a(n)/A007814(a(n)) for 1 <= n <= 2^11, color-coded to show wt(a(n-1)). Wikipedia, Hamming weight Wolfram Research, Numbers in Pascal's triangle EXAMPLE Let wt(n) = A000120(n). 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. a(3) = 3 since wt(2) = 1, we find "1" as first bit of yet unused 3 = "11"_2. a(4) = 4 since wt(3) = 2 = "10"_2, we find "10" as first bits of yet unused 4 = "100"_2. a(5) = 5 since wt(4) = 1, and yet unused 5 = "101"_2 starts with 1. a(6) = 8 since wt(5) = 2 = "10"_2; we see that the yet unused 6 and 7 start with "11"_2, and it isn't until 8 that we have a number that when expressed in binary starts with "10"_2. a(7) = 6 since wt(8) = 1, we can now apply the yet unused 6 = "110"_2 because it starts with 1, etc. MATHEMATICA Nest[Append[#, Block[{k = 1, r = IntegerDigits[DigitCount[#[[-1]], 2, 1], 2]}, While[Nand[FreeQ[#, k], Take[IntegerDigits[k, 2], Length@ r] == r], k++]; k]] & @@ {#, Length@ #} &, {1}, 2^7] PROG (Python) def aupto(n):   alst, used = , {1}   for i in range(2, n+1):     binprev = bin(alst[-1])[2:]     binwt = binprev.count("1")     lsbs, extra = 0, 0     while binwt + extra in used:       lsbs += 1       binwt *= 2       for extra in range(2**lsbs):         if binwt + extra not in used: break     alst.append(binwt+extra); used.add(binwt+extra)   return alst    # use alst[n-1] for a(n) print(aupto(68)) # Michael S. Branicky, Dec 16 2020 CROSSREFS Cf. A000120, A338209, A339607. Sequence in context: A272616 A085176 A354125 * A192179 A118462 A219360 Adjacent sequences:  A339021 A339022 A339023 * A339025 A339026 A339027 KEYWORD nonn,base,easy AUTHOR Michael De Vlieger, Dec 16 2020 STATUS approved

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Last modified July 1 03:30 EDT 2022. Contains 354947 sequences. (Running on oeis4.)