A338409 a(n) is the number of nodes with depth of n in a binary tree defined as: root = 1 and a child (C) of a node (N) is such that A338215(C) = N. For nodes with two children, the smaller child is assigned as the left child and the bigger one as the right child. A child of a one-child node is assigned as the left child.

1, 2, 2, 2, 1, 1, 2, 2, 2, 2, 3, 3, 2, 2, 2, 3, 2, 3, 4, 3, 4, 3, 4, 4, 4, 4, 6, 6, 5, 6, 4, 4, 6, 7, 7, 6, 7, 6, 5, 4, 6, 7, 8, 8, 8, 8, 10, 8, 8, 8, 9, 10, 8, 9, 11, 13, 11, 9, 12, 11, 10, 11, 11, 11, 13, 11, 14, 14, 13, 15, 17, 15, 16, 16, 16, 14, 14, 14

Offset

0,2

Links

Table of n, a(n) for n=0..77.

Example

The binary tree, read from left to right in the order of increasing depth n, is the positive integer sequence A000027. The first 67 numbers are shown in the figure below.
    1
  (2)\_3
     (4)\_5
         6 \_(7)
        8
       9
    (10)\_11
         12 \___________13
        14            (15)
       16 \______17
     (18)\_19   20
          21   22 \_(23)
         24   25
       (26)  27
            28 \______29
           30 \_(31) 32
          33        34
         35        36 \_____________________37
       (38)       39                       40 \_(41)
                 42 \______43             44
                45        46 \______47  (48)
               49        50        51
              52 \_(53) 54        55
            (56)       57        58 \_(59)
                      60 \_(61) 62   63
                     64        65   66 \_67
All left children except 2 are composite numbers and all prime numbers are right children.

Prog

(Python)
from sympy import primepi
def depth(k):
    d = 0
    while k > 1:
        k -= primepi(k)
        k += primepi(k)
        d += 1
    return d
m = 1
for n in range (0, 101):
    a = 0
    while depth(m + a) == n:
        a += 1
    print(a)
    m += a

Crossrefs

Cf. A000027, A062298, A095117, A338215, A338237, A338260.

Sequence in context: A214774 A320107 A190321 * A238890 A266968 A237593

Adjacent sequences: A338406 A338407 A338408 * A338410 A338411 A338412

Keyword

nonn

Author

Ya-Ping Lu, Oct 24 2020

Status

approved