OFFSET
1,2
COMMENTS
There is a strong conjecture that A336957 consists exactly of 1, 2, and all numbers with at least two different prime factors. [The only uncertainty is whether all numbers with at least two prime factors appear.]
The terms in A000961 greater than 2 are definitely missing from A336957, so A336957 is obviously not a permutation of the positive integers.
The present sequence is obtained by inserting the missing prime powers q = p^k, p >= 2, k >= 1, in their natural positions. More precisely, let the terms of A336957 be [W(i), i >= 1].
Between W(i) and W(i+1) we insert, in order, any prime powers q < W(i+1) which are not yet in the new sequence and satisfy gcd(q, W(i)) > 1 and gcd(q, W(i-1) = 1.
It is conjectured that this is a permutation of the positive integers.
LINKS
N. J. A. Sloane, Table of n, a(n) for n = 1..5000
EXAMPLE
Suppose n = 4.
The first 5 terms of A336957 are 1,2,6,15,35. The first 7 terms of the present sequence are 1, 2, 4, 6, 3, 9, 15. To see what comes after a(7) = W(4) = 15, we look at the missing prime powers less than W(5) = 35, which are 5, 7, 8, 11, 13, 16, 17, 19, 23, 25, 27, 29, 31. Just two terms, 5 and 25, have a common factor with 15 and are relatively prime to W(3) = 6, so they are adjoined to the sequence.
In short, we adjoin any missing prime powers which are less than W(n+1), have a common factor with W(n), and are relatively prime to W(n-1). We insert them immediately after W(n).
CROSSREFS
KEYWORD
nonn
AUTHOR
Scott R. Shannon and N. J. A. Sloane, Oct 17 2020
STATUS
approved