A290183 - OEIS

%I #48 Jun 13 2021 03:23:39

%S 59,331,163,79,37,17,724777,941593,3578683,73397147,52457707,26228849,

%T 2600553509,1300276753

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

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

%C 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 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 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 Michel Marcus and Peter Munn, <a href="/A290183/a290183.gp.txt">PARI script</a>

%F 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 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 The "tree of primes" defined above starts:

%e                             Root

%e                +----------------------------+

%e                2                            3

%e                |                            |

%e                5                            7

%e        +--------------+                 +-------+

%e       11             13                17      19

%e        |         +-----------+          |    +----+

%e       23        29          31         37   41   43

%e     +----+    +----+    +----+----+     |    |    |

%e    47   53   59   61   67   71   73    79   83   89

%e The subtree consisting of the node labeled 17 and all 5 of its descendants is:

%e       17

%e        |

%e       37

%e        |

%e       79

%e        |

%e      163

%e        |

%e      331

%e        |

%e      673

%e From _Peter Munn_, Nov 22 2018: (Start)

%e The subtree consisting of the node labeled with a(7) = 941593 and its descendants is:

%e              941593

%e         +------+--------+

%e      1883191         1883197

%e         |               |

%e      3766387         3766397

%e                         |

%e                      7532813

%e       +-----------------+-----------------+

%e   15065627          15065639           15065641

%e       |           +-----------+           |

%e   30131267    30131279    30131281     30131291

%e                                           |

%e                                        60262597

%e                                           |

%e                                       120525217

%e (End)

%o (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 (PARI) ntimes(p, n) = for (k=1, n, p = precprime(p>>1)); p;

%o 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 Cf. A080192 is a list of leaf nodes.

%Y Cf. A055377, A059786, A059788, and A102820 are associated with the parent/child node relationship.

%Y Cf. A000040, A055496, A151800.

%K nonn,more

%O 0,1

%A _Peter Munn_, Jul 23 2017

%E Candidate for a(8) confirmed by _Michel Marcus_, Nov 14 2018

%E a(9)-a(13) from _Michel Marcus_, Jan 02 2019