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