cp's OEIS Frontend

A341742 Nodes read by depth in a binary tree defined as: Root = 1; an even node N has a left child N + 1 if N + 1 is not a prime, and an odd node N has a left child sqrt(N + 2) if sqrt(N + 2) is a prime; the right child of a node N is 2*N.

%I A341742 #15 Mar 18 2021 05:46:26
%S A341742 1,2,4,8,9,16,18,32,36,33,64,72,66,65,128,144,132,130,129,256,145,288,
%T A341742 133,264,260,258,512,290,289,576,266,265,528,261,520,259,516,513,1024,
%U A341742 291,580,578,1152,267,532,530,529,1056,522,1040,518,517,1032,1026
%N A341742 Nodes read by depth in a binary tree defined as: Root = 1; an even node N has a left child N + 1 if N + 1 is not a prime, and an odd node N has a left child sqrt(N + 2) if sqrt(N + 2) is a prime; the right child of a node N is 2*N.
%C A341742 Let d be the depth of a node N in the binary tree and f be the map of A340801. The d-th iteration of map A340801 on N gives 1, or f^d(N) = 1.
%C A341742 If Conjectures 1 and 2 made in A340801 hold, the sequence contains all positive integers and each integer appears once in the sequence.
%C A341742 The first odd prime does not appear until d reaches 30 and the first five odd primes appearing in the sequence are:
%C A341742         n     a(n)     d
%C A341742   -------    -----    --
%C A341742    140735     4099    30
%C A341742    151872     1543    31
%C A341742   1574120     8689    36
%C A341742   1841645     2917    36
%C A341742   2111465    32771    36
%C A341742 The first two odd primes less than 100 appear in the binary tree are 17 at d = 4426 and 71 at d = 4421.
%e A341742 The binary tree for depths up to 9 is given below.
%e A341742   1
%e A341742    \
%e A341742     2
%e A341742      \
%e A341742       4
%e A341742        \
%e A341742         8
%e A341742      /    \
%e A341742    9       16
%e A341742     \        \
%e A341742     18        32
%e A341742      \       /  \
%e A341742      36    33    64
%e A341742       \     \    / \
%e A341742       72    66  65  128
%e A341742        \     \   \   / \
%e A341742       144   132 130 129 256
%e A341742       / \   / \   \   \   \
%e A341742    145 288 133 264 260 258 512
%o A341742 (Python)
%o A341742 from sympy import isprime
%o A341742 from math import sqrt
%o A341742 def children(N):
%o A341742     C = []
%o A341742     if N%2 == 0:
%o A341742         if isprime(N + 1) == 0: C.append(N+1)
%o A341742     else:
%o A341742         p1 = sqrt(N + 2.0); p2 = int(p1 + 0.5)
%o A341742         if p2**2 == N + 2 and isprime(p2) == 1: C.append(p2)
%o A341742     C.append(2*N)
%o A341742     return C
%o A341742 L_last = [1]; print(L_last)
%o A341742 for d in range(1, 18):
%o A341742     L_1 = []
%o A341742     for i in range(0, len(L_last)):
%o A341742         C_i = children(L_last[i])
%o A341742         for j in range(0, len(C_i)): L_1.append(C_i[j])
%o A341742     print(L_1); L_last = L_1
%Y A341742 Cf. A340801, A006577, A340008, A339991, A340419.
%K A341742 nonn
%O A341742 1,2
%A A341742 _Ya-Ping Lu_, Feb 18 2021