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In other words if a(n-1) is a novel term, a(n) is the product of all primes < Gpf(a(n-1)) which do not divide a(n-1), else if a(n-1) seen k times already, up to and including itself then a(n)=k*a(n-1). All positive integers appear eventually in the sequence, and the first occurrences of primes appear in order.

Every squarefree number appears infinitely many times (consequent to both conditions of the definition), whereas numbers m which are not squarefree can appear only from the second condition, and therefore appear finitely many (at most A000005(m)-1) times; see Example.

For n >= 2, a(n) is the largest part minus the number of distinct parts of the partition having Heinz number n. The Heinz number of a partition [i_1, i_2, ..., i_r] is defined as Product_{j=1..r} (i_j-th prime) (concept used by Alois P. Heinz in A215366 as an encoding of a partition). For example, for the partition [1, 1, 1, 4] we get 2*2*2*7 = 56; a(56) = 4 - #{1,4} = 2. - Emeric Deutsch, Jun 09 2015 [edited by Peter Munn, Apr 09 2024]

a(n) = 1 iff n is prime or n is 1.

Apart from 1-terms, the other duplicated terms are: a(24) = a(27), a(120) = a(125), a(140) = a(147), a(528) = a(539), etc, whose positions are listed by A293893 and A293894. - Antti Karttunen, Nov 01 2017

It follows from the definition that the sequence is infinite, and that the records (outside of the first 7 terms) are all primorial numbers, meaning that it grows very quickly.

When there are no primes less than the greatest prime factor of a(n-1) which do not divide a(n-1) then m is the least novel multiple of 1, the empty product, and therefore a(n) = u, the least unused number in the sequence so far. The only way a prime can enter the sequence is as u. When a(n-1) = prime(k), a(n) is A002110(k-1), and any primorial term is followed by u. Thus: prime —> primorial —> u.

Sequence is a permutation of the positive integers since by the definition no number appears more than once and m = 1 eventually introduces any number not already placed by the first part of the definition (m > 1).

a(n+1) is a multiple of A083720(a(n)).

This sequence has similarities with A175343; here we consider prime factors of consecutive terms, there ones in binary expansions of consecutive terms.

This sequence is a self-inverse permutation of the positive integers.

The n-th term is a multiple of A083720(n).

a(n) depends upon if rad(j) = A002110(k) for some k (equivalently A083720(j) = 1), or not. If so a(n) is least novel m such that rad(m*j) = A002110(k+1). Otherwise a(n) = least novel m such that rad(m*j) = A002110(A000720(q)), where q = gpf(j).

Put otherwise, if p = nextprime(q), and A = A083720, then for n > 1 if A(j) = 1, a(n) is the least novel p-smooth number divisible by p, and if A(j) = w > 1, a(n) is the least novel q-smooth number divisible by w.

If j is a term in A002110, a(n) = smallest prime which has not yet appeared in the sequence (e.g., 1-->2, 2-->3, 6-->5, 30-->7, 210-->11, and so on).

Primes are in order and if p is prime and p|a(n) there is an i <= n such that a(i) = p (no multiple of p appears prior to p). Sequence is conjectured to have "Property S" of A368900. Also, for integers x, y with x < y and rad(x) = rad(y), x appears in the sequence before y. Conjecture: Sequence is a permutation of the positive integers which preserves the above mentioned properties of A000027.

Following j = a(n-1), a term in A005932, a(n) is the smallest prime not already listed. Otherwise a(n) = smallest novel product of powers of non divisor primes of j; a number of the form: Product_{i = 0..k} p_i^e_i; p_i a prime < q = Gpf(j) which does not divide j, e_i >= 0, k = the number of primes p_i < q which do not divide j.

Adjacent terms are coprime and the greedy algorithm implied by the definition forces naked prime p to appear in advance of any multiple m*p of p; m >1.

Prime powers enter the sequence early, consequent to j having a single non divisor prime. A power of 3 is always followed by a power of 2.

Conjectures:

(i) A permutation of the positive integers in which the primes appear in order.

(ii)The sequence obeys Selcoe's theorem (see A280864) regarding numbers that have the same squarefree kernel, namely: Construct a sequence S_r = { m*r : rad(m) | r } = { k : rad(k) = r }, squarefree r. Terms w in S_r appear in this sequence in order. This is to say, for example, that for r = 6, terms in A033845 = {6, 12, 18, 24, 36, 48, 54, ...} appear in order.

When n is prime, n = largest prime dividing n; hence a(n) is the sum of all primes less than n = A034387(n)-n. a(n) = SUM{p such that p is in A000040 AND NOT(p|n) AND p < A006530(n)}. - Jonathan Vos Post, Mar 08 2006

The zeros give A055932: All prime divisors are consecutive primes starting at 2. - Robert G. Wilson v, May 01 2006