cp's OEIS Frontend

From Antti Karttunen, Apr 07 2020: (Start)

Also the least number of iterations of nondeterministic map k -> k-(k/p) needed to reach a power of 2, when any prime factor p of k can be used. The minimal length path to the nearest power of 2 (= 2^A064415(n)) is realized whenever one uses any of the A005087(k) distinct odd prime factors of the current k, at any step of the process. For example, this could be done by iterating with the map k -> k-(k/A078701(k)), i.e., by using the least odd prime factor of k (instead of the largest prime).

Proof: Viewing the prime factorization of changing k as a multiset ("bag") of primes, we see that liquefying any odd prime p with step p -> (p-1) brings at least one more 2 to the bag, while applying p -> (p-1) to any 2 just removes it from the bag, but gives nothing back. Thus the largest (and thus also the nearest) power of 2 is reached by eliminating - step by step - all odd primes from the bag, but none of 2's, and it doesn't matter in which order this is done.

The above implies also that the sequence is totally additive, which also follows because both A064097 and A064415 are. That A064097(n) = A329697(n) + A054725(n) for all n > 1 can be also seen by comparing the initial conditions and the recursion formulas of these three sequences.

For any n, A333787(n) is either the nearest power of 2 reached (= 2^A064415(n)), or occurs on some of the paths from n to there.

(End)

A003401 gives the numbers k where a(k) = A005087(k). See also A336477. - Antti Karttunen, Mar 16 2021

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.

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.

Applying map k -> (p-1)*(k/p) to any term k on any row n > 1, where p is any prime factor of k, gives one of the terms on preceding row n-1.

Any 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 A064097 is completely additive.

A001221(k) gives the number of terms on the row above that are immediate descendants of k.

A067513(k) gives the number of terms on the row below that lead to k.

Restricted growth sequence transform of the ordered pair [A329697(n), A336158(n)].

For all i, j:

A336470(i) = A336470(j) => a(i) = a(j)

a(i) = a(j) => A336396(i) = A336396(j),

a(i) = a(j) => A336469(i) = A336469(j) => A336477(i) = A336477(j).

This sequence has an ability to see where the terms of A003401 are, as they are the indices of zeros in A336469. Specifically, they are numbers k that satisfy the condition A329697(k) = A001221(A336158(k)), i.e., numbers for which A329697(k) is equal to the number of distinct prime divisors of the odd part of k. See also comments in array A334100.

Array is read by descending antidiagonals with (n,k) = (0,0), (0,1), (1,0), (0,2), (1,1), (2,0), ... where A(n,k) is the (k+1)-th solution x to A331410(x) = n. The row indexing (n) starts from 0, and column indexing (k) also from 0.

For any odd prime p that appears on row n, p+1 appears 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 A331410 is completely additive.