A338237 -id:A338237 - cp's OEIS frontend

A348907 If n is prime, a(n) = n, else a(n) = a(n-pi(n)), n >= 2; where pi is the prime counting function A000720.

2, 3, 2, 5, 3, 7, 2, 5, 3, 11, 7, 13, 2, 5, 3, 17, 11, 19, 7, 13, 2, 23, 5, 3, 17, 11, 19, 29, 7, 31, 13, 2, 23, 5, 3, 37, 17, 11, 19, 41, 29, 43, 7, 31, 13, 47, 2, 23, 5, 3, 37, 53, 17, 11, 19, 41, 29, 59, 43, 61, 7, 31, 13, 47, 2, 67, 23, 5, 3, 71, 37, 73, 53, 17, 11

Offset: 2

David James Sycamore, Nov 03 2021

A fractal sequence in which every term is prime. The proper subsequence a(k), for composite numbers k = 4,6,8,9... is identical to the original, and the records subsequence is A000040.

Regarding this sequence as an irregular triangle T(m,j) where the rows m terminate with 2 exhibits row length A338237(m). In such rows m, we have a permutation of the least A338237(m) primes. - Michael De Vlieger, Nov 04 2021

			2 is prime so a(2) = 2.
3 is prime so a(3) = 3.
4 is not prime so a(4) = a(4-pi(4)) = 2.
5 is prime so a(5) = 5.
6 is composite so a(6) = a(6-pi(6)) = 3.
From _Michael De Vlieger_, Nov 04 2021: (Start)
Table showing pi(a(n)) for the first rows m of this sequence seen as an irregular triangle T(m,j). "New" primes introduced for prime n are shown in parentheses:
m\j   1   2   3   4   5   6   7   8   9  10  11   A338237(m)
------------------------------------------------------------
1:   (1)                                                1
2:   (2)  1                                             2
3:   (3)  2  (4)  1                                     4
4:    3   2  (5)  4  (6)  1                             6
5:    3   2  (7)  5  (8)  4   6   1                     8
6:   (9)  3   2   7   5   8 (10)  4 (11)  6   1        11
... (End)

Michael De Vlieger, Table of n, a(n) for n = 2..10238 (as an irregular triangle, rows 1 <= n <= 35 flattened).
Michael De Vlieger, Log-log scatterplot of a(n), for n=1..2^16.

Cf. A000040, A002808, A000720, A010051, A338237.

a[n_]:=If[PrimeQ@n,n,a[n-PrimePi@n]];Array[a,75,2] (* Giorgos Kalogeropoulos, Nov 03 2021 *)

a(n) = if (isprime(n), n, a(n-primepi(n))); \\ Michel Marcus, Nov 03 2021

from sympy import isprime
def aupton(nn):
    alst, primepi = [], 0
    for n in range(2, nn+1):
        if isprime(n): an, primepi = n, primepi + 1
        else: an = alst[n - primepi - 2]
        alst.append(an)
    return alst
print(aupton(76)) # Michael S. Branicky, Nov 04 2021

More terms from Michel Marcus, Nov 03 2021

A349191 a(n) = A000720(A348907(n+1)).

Original entry on oeis.org

1, 2, 1, 3, 2, 4, 1, 3, 2, 5, 4, 6, 1, 3, 2, 7, 5, 8, 4, 6, 1, 9, 3, 2, 7, 5, 8, 10, 4, 11, 6, 1, 9, 3, 2, 12, 7, 5, 8, 13, 10, 14, 4, 11, 6, 15, 1, 9, 3, 2, 12, 16, 7, 5, 8, 13, 10, 17, 14, 18, 4, 11, 6, 15, 1, 19, 9, 3, 2, 20, 12, 21, 16, 7, 5, 8, 13, 22, 10

Offset: 1

Michael De Vlieger, Nov 09 2021

Regarding this sequence as an irregular triangle T(m,j) where the rows m terminate with 1 exhibits row length A338237(m). In such rows m, we have a permutation of the range of natural numbers 1..A338237(m).

Records are the natural numbers.

			Table showing a(n) for the first rows m of this sequence seen as an irregular triangle T(m,j). "New" numbers introduced for prime (n+1) are shown in parentheses:
  m\j   1   2   3   4   5   6   7   8   9  10  11   A338237(m)
  ------------------------------------------------------------
  1:   (1)                                                1
  2:   (2)  1                                             2
  3:   (3)  2  (4)  1                                     4
  4:    3   2  (5)  4  (6)  1                             6
  5:    3   2  (7)  5  (8)  4   6   1                     8
  6:   (9)  3   2   7   5   8 (10)  4 (11)  6   1        11
  ...

Michael De Vlieger, Table of n, a(n) for n = 1..10237 (as an irregular triangle, rows 1 <= n <= 36, flattened)
Michael De Vlieger, Log-log scatterplot of a(n) for 1 <= n <= 11636, showing 36 rows if read as an irregular table.

Cf. A000027, A000040, A000720, A338237, A348907, A349192.

c = 0; 1 + Reap[Do[Set[a[i], If[PrimeQ[i], i; c++, a[i - c]] ]; Sow[a[i]], {i, 2, 2^24}] ][[-1, -1]]

A349192 Irregular triangle T(m,k) = inverse permutation of S(m,k) = A349191 read as an irregular triangle.

Original entry on oeis.org

1, 2, 1, 4, 2, 1, 3, 6, 2, 1, 4, 3, 5, 8, 2, 1, 6, 4, 7, 3, 5, 11, 3, 2, 8, 5, 10, 4, 6, 1, 7, 9, 15, 3, 2, 11, 6, 13, 5, 7, 1, 9, 12, 4, 8, 10, 14, 18, 3, 2, 14, 7, 16, 6, 8, 1, 10, 15, 4, 9, 12, 17, 5, 11, 13, 24, 4, 3, 19, 10, 21, 9, 11, 2, 14, 20, 6, 12, 16

Offset: 1

Michael De Vlieger, Nov 09 2021

We find k at S(m,k) where S is A349191 read as an irregular triangle. Alternatively, we find prime(k) at U(m,k) where U is A348907 read as an irregular triangle.

			First rows of T(m,k):
  m\k   1   2   3   4   5   6   7   8   9  10  11
  -----------------------------------------------
  1:    1
  2:    2   1
  3:    4   2   1   3
  4:    6   2   1   4   3   5
  5:    8   2   1   6   4   7   3   5
  6:   11   3   2   8   5  10   4   6   1   7   9
  ...

Michael De Vlieger, Table of n, a(n) for n = 1..10237 (rows 1 <= n <= 35, flattened)
Michael De Vlieger, Log-log scatterplot of T(m,k) 1 <= m <= 36.

Cf. A000027, A000040, A000720, A338237, A348907, A349191.

c = 0; Flatten@ Map[Table[If[k == 1, Length[#] + 1, FirstPosition[#, k - 1][[1]]], {k, If[IntegerQ@ #, # + 1, 1] &@ Max[#]}] &, {{}}~Join~Most@ SplitBy[Reap[Do[Set[a[i], If[PrimeQ[i], i; c++, a[i - c]]]; Sow[a[i]], {i, 2, 100}]][[-1, -1]], # == 0 &][[2 ;; -1 ;; 2]]]

Row lengths are in A338237.

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.

Original entry on oeis.org

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

Ya-Ping Lu, Oct 24 2020

nonn

			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.

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

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

cp's OEIS Frontend

A348907 If n is prime, a(n) = n, else a(n) = a(n-pi(n)), n >= 2; where pi is the prime counting function A000720.

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A349191 a(n) = A000720(A348907(n+1)).

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A349192 Irregular triangle T(m,k) = inverse permutation of S(m,k) = A349191 read as an irregular triangle.

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