cp's OEIS Frontend

Define binary weight wt(n) as A000120(n), the number of 1s in the binary expansion of n. Let w = A000120(a(n-1)) the binary weight of the previous term. In other words, a(n) is the least m not already in the sequence such that m mod 2^k = w, where k = floor(1 + log_2 w).

Likely a permutation of the natural numbers.

The numbers m = 2^k with 0 <= k <= 3 appear at indices {1, 3, 11, 222}. The term 16 has not appeared for n <= 2^14 and may not until n approaches 2^16.

The numbers m = (2^k + 1) appear at indices {2, 4, 12, 223, ...}. The numbers m = 2^k or (2^k + 1) require n approximately equal to 2^m in order to appear in the sequence.

The numbers m = (2^k - 1) with 1 <= k <= 14 appear at indices {1, 2, 8, 10, 23, 43, 130, 278, 447, 758, 1390, 2525, 4719, 9333}, respectively.

The plot exhibits dendritic streams of residues r (mod 2^k). We can identify coordinates (x, y) = (n, a(n)) on the plot where the streams branch.

The branches of the tree in the plot contain m congruent to r (mod 2^k), where r is a term (except the last term) in row (k-1) of A049773.

Given 2^14 terms of this sequence, we see 2 or 3 successive invocations of w, otherwise, w appears just once before a different value succeeds it in the next term.

2^4 appears at index 47201. - Michael S. Branicky, Dec 16 2020

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

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

a(n) = 10^k occurs before n = 10^k for 0 < k <= 5.

Conjecture: permutation of the nonnegative numbers.

A decimal version of A339607, a permutation of the integers having to do with binary weight, but instead using digital root rather than digit sum. The plots of these sequences have similar features.