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A290183 a(n) is least p to label a node of height n in the rooted tree which has the primes (A000040) as labels for its nonroot nodes, and the parent of the node labeled p is the node labeled with the greatest prime less than p/2 (or the root if there is no such node).

%I A290183 #48 Jun 13 2021 03:23:39
%S A290183 59,331,163,79,37,17,724777,941593,3578683,73397147,52457707,26228849,
%T A290183 2600553509,1300276753
%N A290183 a(n) is least p to label a node of height n in the rooted tree which has the primes (A000040) as labels for its nonroot nodes, and the parent of the node labeled p is the node labeled with the greatest prime less than p/2 (or the root if there is no such node).
%C A290183 The parent of the node labeled p is the adjacent node through which it is connected to the root.
%C A290183 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.
%C A290183 If the set of heights of nodes has a greatest finite value, k, this sequence is finite with k+1 contiguous defined terms.
%C A290183 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.
%C A290183 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.
%C A290183 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.
%H A290183 Michel Marcus and Peter Munn, <a href="/A290183/a290183.gp.txt">PARI script</a>
%F A290183 a(n) = min (A055377^(n+1) \ A055377^(n+2)), where A055377^(i) denotes the image set of N after i applications of A055377. - clarified following suggestion by _Michel Marcus_, Nov 13 2018
%e A290183 The only prime between 2 * 331 = 662 and 2 * nextprime(331) = 2 * 337 = 674 is 673, so the node labeled 331 is parent only to the node labeled 673. There are no primes between 2 * 673 = 1346 and 2 * nextprime(673) = 2 * 677 = 1354, so the node labeled 673 is a leaf. Thus the node labeled 331 has height 1. It is the least such prime, so a(1) = 331.
%e A290183 The "tree of primes" defined above starts:
%e A290183                             Root
%e A290183                +----------------------------+
%e A290183                2                            3
%e A290183                |                            |
%e A290183                5                            7
%e A290183        +--------------+                 +-------+
%e A290183       11             13                17      19
%e A290183        |         +-----------+          |    +----+
%e A290183       23        29          31         37   41   43
%e A290183     +----+    +----+    +----+----+     |    |    |
%e A290183    47   53   59   61   67   71   73    79   83   89
%e A290183 The subtree consisting of the node labeled 17 and all 5 of its descendants is:
%e A290183       17
%e A290183        |
%e A290183       37
%e A290183        |
%e A290183       79
%e A290183        |
%e A290183      163
%e A290183        |
%e A290183      331
%e A290183        |
%e A290183      673
%e A290183 From _Peter Munn_, Nov 22 2018: (Start)
%e A290183 The subtree consisting of the node labeled with a(7) = 941593 and its descendants is:
%e A290183              941593
%e A290183         +------+--------+
%e A290183      1883191         1883197
%e A290183         |               |
%e A290183      3766387         3766397
%e A290183                         |
%e A290183                      7532813
%e A290183       +-----------------+-----------------+
%e A290183   15065627          15065639           15065641
%e A290183       |           +-----------+           |
%e A290183   30131267    30131279    30131281     30131291
%e A290183                                           |
%e A290183                                        60262597
%e A290183                                           |
%e A290183                                       120525217
%e A290183 (End)
%o A290183 (PARI) lista(nbp) = {my(v = primes(nbp), nv, x); for (n=0, oo, nv = Set(apply(x->precprime(x >> 1), v)); x = vecmin(setminus(v, nv)); if (x > vecmax(nv), break); print1(x, ", "); v = nv;);} /* use nbp=6*10^6 for 7 terms and nbp=2*10^7 for 8 terms */ \\ _Michel Marcus_, Nov 19 2018
%o A290183 (PARI) ntimes(p, n) = for (k=1, n, p = precprime(p>>1)); p;
%o A290183 a(n) = {p = 2; x = ntimes(p, n); p = nextprime(p+1); q = 5; y = ntimes(q, n+1); q = nextprime(q+1); for (k= 1, oo, if (y != x, return (x)); until((ny = ntimes(q, n+1)) != x, q = nextprime(q+1)); until((nx = ntimes(p, n)) != x, p = nextprime(p+1)); x = nx; y = ny;);} \\ _Michel Marcus_, Dec 08 2018
%Y A290183 Cf. A080192 is a list of leaf nodes.
%Y A290183 Cf. A055377, A059786, A059788, and A102820 are associated with the parent/child node relationship.
%Y A290183 Cf. A000040, A055496, A151800.
%K A290183 nonn,more
%O A290183 0,1
%A A290183 _Peter Munn_, Jul 23 2017
%E A290183 Candidate for a(8) confirmed by _Michel Marcus_, Nov 14 2018
%E A290183 a(9)-a(13) from _Michel Marcus_, Jan 02 2019