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Semiprime analog of A059785 a(n+1)=prevprime(a(n)^2), with a(1) = 2. With that, of course, there's always a prime between n and 2n, so a(n) < 2^n. See also A055496 a(1) = 2; a(n) is smallest prime > 2*a(n-1). The obverse of this is A118909 a(1) = 4; a(n) is least semiprime > a(n-1)^2.

a(9), which is too large to be included, is equal to a(8)^2-3. - Giovanni Resta, Jun 16 2016

Conjectures: (row 1) = A055496, (column 1) = A193507, and for each row r(k), the limit of r(k)/2^k exists. For rows 1 to 4, the respective limits are 1.569985..., 2.677285..., 8.230592..., 10.709142...; see Franklin T. Adams-Watters's comment at A055496.

The above conjecture row 1 = A055496 is true; additionally, row 2 = A065545; row 3 = A065546; the first 5 terms of row 6 are a contiguous subsequence of A064934; and column 1 = A194598. - Bob Selcoe, Oct 27 2015; corrected by Peter Munn, Jul 30 2017

The conjecture for column 1 is true iff A194598 and A193507 are equivalent. Is this the case? - Bob Selcoe, Oct 29 2015

Column 1 diverges from A193507 at A(14,1) = 113, a prime not in A193507. 113 is in column 1 as it does not follow a prime in a row: 107 follows 53 and 127 follows 59, the next prime after 53. - Peter Munn, Jul 30 2017

There is no common term with A055496? - Zak Seidov, Feb 04 2016

Correct, there is no common term with A055496. - Flávio V. Fernandes, Apr 10 2021

The parent of the node labeled p is the adjacent node through which it is connected to the root.

The "tree of primes" defined above relates to many older sequences. The node labeled A055377(n) is parent of the node labeled n. The node labeled prime(k) has A102820(k) child nodes and unless it has no child nodes, these are labeled with the primes from A059786(k) to A059788(k+1). The leaf node labels are A080192. The nodes of depth m are those with labels in the interval [A055496(m), A055496(m+1)). The full tree may be defined using A000040 read as a table with row lengths given by A102820 prefixed by 2.

If the set of heights of nodes has a greatest finite value, k, this sequence is finite with k+1 contiguous defined terms.

That said, the author's initial assessment is that occurrence of height n nodes will have similarities to occurrence of least primes of prime k-tuples, namely: (1) labels of nodes of height n will occur almost as though at random intervals amongst the primes; (2) for any n, the apparent odds against a prime p being such a label will not be greater than polynomial in log(p); and thus (3) a(n) plausibly exists for all n.

Some initial empirical observation suggests nodes of height n+1 may occur something like 5 to 10 times less frequently than those of height n.

Terms a(1) to a(5) come from the subtree consisting of the node labeled 17 and its descendants, as depicted in the example section below. This implies 4 consecutive negative first differences, which may be rare later in the sequence.

Exponent 3 analog of A059785.

Obverse of this is A051254.

Conjectures: (Strong) Let x,y be 2 positive integers and define a(n) as a(1)=1, a(n)=x*a(n-1) if a(n-1) is prime, a(n)=a(n-1)+y otherwise; then lim_{n->oo} log(a(n))/sqrt(n) = C(x,y) exists. (Weak) log(a(n))/sqrt(n) is bounded.

Numbers k(n) are given in A105121.

a(n) appears to tend toward C*A055496(n), C~ 0.992521946129820000. - Bill McEachen, Feb 21 2022

Semiprime analog of A055496 a(1) = 2; a(n) is smallest prime > 2*a(n-1). See also A059785 a(n+1)=prevprime(a(n)^2), with a(1) = 2. With that, of course, there's always a prime between n and 2n, so a(n) < 2^n. The obverse of this is A118908 a(1) = 4; a(n) is greatest semiprime < a(n-1)^2.