cp's OEIS Frontend

Fixed points of the Doudna sequence: A005940(a(n)) = A005941(a(n)) = a(n). - Reinhard Zumkeller, Aug 23 2006

Subsequence of A103969. - R. J. Mathar, Mar 06 2010

Question: Is there a simple proof that A005940(c) = c would never allow an odd composite c as a solution? See also my comments in A163511 and in A335431 concerning similar problems, also A364551 and A364576. - Antti Karttunen, Jul 28 & Aug 11 2023

Let's define "property S" for sequences as follows: If s is any sequence of positive natural numbers, normalized to begin with offset 1, then it satisfies the S-property if LCM-transform(s) is equal to the sequence obtained by applying A014963 to sequence s, or in other words, when for all n >= 1, lcm {s(1)..s(n)} / lcm {s(1)..s(n-1)} = A014963(s(n)). This holds if and only if, for all n >= 1, when, either (case A): s(n) is of the form p^k, p prime, then gcd(s(n), lcm {s(1)..s(n-1)}) must be equal to p^(k-1), or (case B): when s(n) is not a prime power, then gcd(s(n), lcm {s(1)..s(n-1)}) must be equal to s(n). Together the cases (A) and (B) reduce to the condition that each prime power should appear in s before any of its multiples do.

Clearly the Doudna-sequence satisfies the property by the way of its construction, as do many of its variants like A356867 (see A369060).

Also, for any base-2 related permutation b that keeps all the numbers of range [2^k, 2^(1+k)[ in the same range, i.e., if for all n >= 1, A000523(b(n)) = A000523(n), then the above property is automatically satisfied.

Furthermore, because in Doudna-sequence no multiple of any term is located on the same row as the term itself (see the tree-illustration in A005940), it follows that any composition of A005940 with any such base-2 related permutation as mentioned above also automatically satisfies the S-property, for example, the permutations A163511, A243353, A253563, A253565, A366260, A366263 and A366275.

Note: Like A005940 itself, also this sequence might be more logical with the starting offset 0 instead of 1, to better align with the underlying mapping from the binary expansion of n to the prime factorization. - Antti Karttunen, Jan 24 2024

This can be viewed as yet another "entanglement permutation" where the two complementary pairs to be interwoven together are even and odd numbers (A005843/A005408) which are entangled with the complementary pair even numbers (taken straight) and odd numbers in the order they appear in A003961: (A005843/A003961). Sequence A163511 has almost the same definition, but its domain starts from 0, which results a different permutation.

Numbers k such that A005941(k) <= k.

Sequence A005940(A364542(.)) sorted into ascending order.

If k is a term, then also 2*k is present in this sequence, and vice versa.

Shares with A135764 the property that A001511(n) = k for all terms n on row k and when going downwards in each column, terms grow by doubling.

Sequence A005941(A364560(.)) sorted into ascending order.

A029747 is included as a subsequence, because it gives the known fixed points of map n -> A005940(n).

Differs from A343107 for the first time at a(22) = 25, which term is not present in A343107. On the other hand, 35 is the first term of A343107 that is not present in this sequence.

Odd numbers k such that A005941(k) <= k.

All the odd fixed points of map n -> A005940(n) [and its inverse, map n -> A005941(n)] are included in this sequence. This includes both the known odd fixed points, 1, 3 and 5 (see A029747), and any additional hypothetical odd composites that would satisfy the condition n == A005940(n).

This is a subsequence of A364561, so the comments given in A364564 apply also here.

Sequence A005941(A364548(.)) sorted into ascending order.

If k is a term, then also 2*k is present in this sequence, and vice versa.

A029747 is included as a subsequence, because it gives the known fixed points of map n -> A005940(n).