cp's OEIS Frontend

Or, binary irreducible polynomials, interpreted as binary vectors, then written in base 10.

The numbers {a(n)} are a subset of the set {A206074}. - Thomas Ordowski, Feb 21 2014

2^n - 1 is a term if and only if n = 2 or n is a prime and 2 is a primitive root modulo n. - Jianing Song, May 10 2021

For odd k, k is a term if and only if binary_reverse(k) = A145341((k+1)/2) is. - Joerg Arndt and Jianing Song, May 10 2021

"Encoding" means the number whose binary representation is given by the coefficients of the polynomial, e.g., 13=1101[2] encodes X^3+X^2+1. The product is the usual multiplication of polynomials in GF(2)[X] (or binary multiplication without carry-bits, cf. A048720).

a(n) <= n. [As all terms of the table A061858 are nonnegative]

More formally: the lexicographically earliest injection of natural numbers such that for any n > 1, A061858(a(n), a(n-1)) = 0; a(1) = 1. By necessity also surjective on N (see below for why), thus a bijection.

Less formally:

In this context we say that two positive natural numbers x and y "match", when they will not produce any carries when multiplied in binary system (see the Examples). The purpose of this sequence is with a simple greedy algorithm to form pairs of natural numbers that "match to each other" according to that criterion. Note that each number after 1 will satisfy the matching condition both with its predecessor and its successor.

For the sake of this discussion, we call a natural number n "dense" if the density of 1-bits in its binary representation (cf., e.g., A265917) is over a certain threshold, whose exact value we leave undefined, but can be subjectively gauged. In contrast, we call a number "ethereal" if its base-2 representation consists mostly of zeros. E.g., 258 = 100000010_2 is clearly one of the "ethereals", while 43 = 101011_2, is definitely on the denser side.

When running the algorithm, we note that after a while, for long stretches of time, it mostly matches "dense" numbers with "ethereal" numbers, like 258 and 43, which occur next to each other in the sequence as a(76) and a(77), and also a(49)=31 and a(50)=256, which are the most dense and most ethereal members of their respective binary sizes (see the Example section).

Also, it should be obvious that each number of the form 2^k (terms of A000079, the "super-ethereals") occur as the first representative of the numbers of the same binary length, and any number of the form (2^k)-1 (A000225, "super-dense") comes as the last of the numbers of binary length k.

No matter how dense some number might look to us, there is always a sufficiently ethereal number with which it can be mated (that is, the algorithm is never stuck, because it can always try the next unused super-ethereal 2^k if everything else fails). Moreover, whenever that next 2^k has appeared, it also always immediately picks up from the backlog of (more or less dense) numbers the least unmatched number so far, which proves that no number is left out, and the sequence is indeed a permutation of the natural numbers.

However, certain numbers intuitively feel to be much better matches to each other, like 10 and 12 (cf. Examples), because they are not so distant from each other. We define "good matches" to be such pairs that the binary length (A070939) of the numbers is equal. As 10 and 12 are both four bits long, they are one instance of such a good match. Note that 10 is also a good match with the immediately preceding number in the sequence, 9 = 1001_2.

Sequence A266197 gives the positions of these good matches, and A265748 & A265749 give their first and second members respectively. It is an open question whether the algorithm generates an infinite number of good matches or not.

This is a subsequence of the sequence which gives all such n that A001222(n) = A091222(n).

Note: Start indexing from n=1 if you want just composite numbers. a(0)=1 is the only nonprime, noncomposite in this list.

The first term with three prime divisors is a(11) = 255 = 3*5*17.

The next terms with three prime divisors are

255, 3855, 13107, 21845, 24415, 28527, 30583, 31215, 31611, 31695, 32691, 48059, 56283, 56797, 61935, 65365, 87805, 98005, ...

Of these 24415 (= 5*19*257) is the first one with at least one prime factor that is not a Fermat prime (A019434).

The first term with four prime divisors is a(427) = 65535 = 3*5*17*257.

The first terms which are not multiples of any Fermat prime are: 511, 959, 3647, 4039, 4847, 5371, 7141, 7231, 7679, 7913, 8071, 9179, 12179, ... (511 = 7*73, 959 = 7*137, ...)

a(n) is the difference of n and the number obtained when the prime divisors of n are multiplied together in such a way that the carry-bits from intermediate products are discarded, as in A048720.

All terms are divisible by 4.

Note that a number occurs both here and in A265749 if and only if it is a good match both with its predecessor and the successor in A266195.

Note that a number occurs both here and in A265748 if it is a good match both with its precedent and the successor in A266195.