cp's OEIS Frontend

Array is read by descending antidiagonals with (n,k) = (0,1), (0,2), (1,1), (0,3), (1,2), (2,1), ... where A(n,k) is the k-th solution x to A329697(x) = n. The row indexing (n) starts from 0, and column indexing (k) from 1.

Any odd prime that appears on row n is 1+{some term on row n-1}.

The e-th powers of the terms on row n form a subset of terms on row (e*n). More generally, a product of terms that occur on rows i_1, i_2, ..., i_k can be found at row (i_1 + i_2 + ... + i_k), because A329697 is completely additive.

The binary weight (A000120) of any term on row n is at most 2^n.

Primes p such that p-1 is not a power of two, but for which A171462(p-1) = (p-1-A052126(p-1)) is [a power of 2].

Primes of the form ((2^(2^k))+1)*2^h + 1, where ((2^(2^k))+1) is one of the Fermat primes, A019434, 3, 5, 17, 257, ..., .

Numbers n for which A171462(n) = n-A052126(n) is in A334101.

Numbers k such that A000265(k) is either in A333788 or in A334092.

Each term is either of the form A334092(n)*2^k, for some n >= 1, and k >= 0, or a product of two terms of A334101, whether distinct or not.

Binary weight (A000120) of these terms is always either 2, 3 or 4. It is 2 for those terms that are of the form 9*2^k, 4 for the terms of the form p*q*2^k, where p and q are two distinct Fermat primes (A019434), and 3 for the both terms of the form A334092(n)*2^k, and for the terms of the form (p^2)*(2^k), where p is a Fermat prime > 3.

Numbers k that themselves are not powers of two, but for which A171462(k) = k-A052126(k) is [a power of 2].

Numbers k such that A000265(k) is in A019434.

Squares of these numbers can be found (as a subset) in A334102, and the cubes (as a subset) in A334103.

As the underlying sequence A163511 can be represented as a binary tree, so can be this:

0

|

...................0...................

0 1

0......../ \........2 1......../ \........1

/ \ / \ / \ / \

/ \ / \ / \ / \

/ \ / \ / \ / \

0 3 2 2 1 2 1 2

0 4 3 3 2 3 2 4 1 3 2 3 1 3 2 2

etc.

The nodes at the left edge are all zeros, and their right-hand children give positive integers, A000027.

Each left-hand leaning branch stays constant, because A329697(2n) = A329697(n).

The right-hand leaning branches are not necessarily monotonic. For example, a((2^6)-1) = 2 > 1 = a((2^7)-1), because A000040(7) = 17 is a Fermat prime (but A000040(6) = 13 is not), and therefore the latter is only one step away from a power of 2.

Restricted growth sequence transform of the ordered pair [A329697(n), A331410(n)].

For all i, j:

A324400(i) = A324400(j) => a(i) = a(j),

a(i) = a(j) => A334861(i) = A334861(j),

a(i) = a(j) => A335875(i) = A335875(j),

a(i) = a(j) => A335877(i) = A335877(j),

a(i) = a(j) => A335881(i) = A335881(j).

Numbers n for which A171462(n) = n-A052126(n) is in A334102.

Among the first 2821 terms (terms < 2^31), there are terms with binary weights 2, 3, 4, 5, 6 and 8. For example, 33 is the first term with binary weight 2, and 255 is the first term with binary weight 8.

Squares of A334102 form a subsequence.

Among the first 12193 terms (terms < 2^31), there are terms with binary weights 2 - 16, except no terms with weight 13, 14 or 15. For example, 1025 is the first term with binary weight 2, and 65535 is the first term with binary weight 16.