A340419 -id:A340419 - cp's OEIS frontend

A346063 a(n) = primepi(A039634(prime(n)^2-1)).

2, 1, 2, 2, 4, 3, 1, 5, 1, 6, 4, 3, 6, 4, 7, 14, 6, 10, 7, 37, 23, 25, 28, 18, 21, 22, 67, 24, 9, 46, 11, 19, 62, 12, 40, 24, 2, 27, 6, 91, 11, 31, 20, 1, 36, 203, 69, 25, 2, 80, 16, 48, 155, 18, 1, 326, 7, 20, 109, 365, 8, 39, 9, 240, 438, 2, 16, 154, 10, 17

Offset: 1

Ya-Ping Lu, Jul 03 2021

nonn

This sequence looks at the effect on p^2 - 1 of A039634 with the primes represented by their indices. It seems that primes obtained by iterating the map A039634 on p^2 - 1 never fall into a cycle before reaching 2. Conjecture: Iterating the map k -> a(k) eventually reaches 1. For example, 1 -> 2 -> 1; 5 -> 4 -> 2 -> 1; and 27 -> 67 -> 16 -> 14 -> 4 -> 2 -> 1.

If the conjecture holds, then A339991(n) != -1 and A340419 is a finite sequence.

Ya-Ping Lu, Plot of a(n) vs n for n up to 10000

Cf. A000040, A000720, A039634, A339991, A340008, A340418, A340419.

Array[PrimePi@ FixedPoint[If[EvenQ[#] && # > 2, #/2, If[PrimeQ[#] || (# === 1), #, (# - 1)/2]] &, Prime[#]^2 - 1] &, 70] (* Michael De Vlieger, Jul 06 2021 *)

from sympy import prime, isprime, primepi
def a(n):
    p = prime(n); m = p*p - 1
    while not isprime(m): m = m//2
    return primepi(m)
for n in range(1, 71): print(a(n))

a(n) = A000720(A039634(A000040(n)^2-1)). - Pontus von Brömssen, Jul 03 2021

A340801 a(n) is the image of n under the map f defined as f(n) = n^2 - 2 if n is an odd prime, f(n) = n/2 if n is even, and f(n) = n - 1 otherwise.

Original entry on oeis.org

0, 1, 7, 2, 23, 3, 47, 4, 8, 5, 119, 6, 167, 7, 14, 8, 287, 9, 359, 10, 20, 11, 527, 12, 24, 13, 26, 14, 839, 15, 959, 16, 32, 17, 34, 18, 1367, 19, 38, 20, 1679, 21, 1847, 22, 44, 23, 2207, 24, 48, 25, 50, 26, 2807, 27, 54, 28, 56, 29, 3479, 30, 3719, 31, 62

Offset: 1

Ya-Ping Lu, Jan 21 2021

nonn

Conjecture 1: Iterating map f on an integer n (n > 1) results in a different integer, or f^i(n) != f^j(n) if i != j, where f^i(n) and f^j(n) are the i-th and j-th iterations of map f on n respectively.

Conjecture 2: An integer n eventually reaches 1 when map f is applied to n repeatedly.

Cf. A339991, A340008, A340419, A006577.

a(n) = if (n%2, if (isprime(n), n^2-2, n-1), n/2); \\ Michel Marcus, Jan 22 2021

from sympy import isprime
for n in range(1, 101):
    if isprime(n) == 1 and n != 2: a = n*n - 2
    elif n%2 == 0: a = n/2
    else: a = n - 1
    print(a)

a(2*k+1) = 2*a(2*k) if 2*k+1 is not a prime.

a(2*k+2) = a(2*k) + 1, where k >= 1.

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.

Original entry on oeis.org

1, 2, 4, 8, 9, 16, 18, 32, 36, 33, 64, 72, 66, 65, 128, 144, 132, 130, 129, 256, 145, 288, 133, 264, 260, 258, 512, 290, 289, 576, 266, 265, 528, 261, 520, 259, 516, 513, 1024, 291, 580, 578, 1152, 267, 532, 530, 529, 1056, 522, 1040, 518, 517, 1032, 1026

Offset: 1

Ya-Ping Lu, Feb 18 2021

nonn

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.
If Conjectures 1 and 2 made in A340801 hold, the sequence contains all positive integers and each integer appears once in the sequence.
The first odd prime does not appear until d reaches 30 and the first five odd primes appearing in the sequence are:
        n     a(n)     d
  -------    -----    --
   140735     4099    30
   151872     1543    31
  1574120     8689    36
  1841645     2917    36
  2111465    32771    36
The first two odd primes less than 100 appear in the binary tree are 17 at d = 4426 and 71 at d = 4421.

			The binary tree for depths up to 9 is given below.
  1
   \
    2
     \
      4
       \
        8
     /    \
   9       16
    \        \
    18        32
     \       /  \
     36    33    64
      \     \    / \
      72    66  65  128
       \     \   \   / \
      144   132 130 129 256
      / \   / \   \   \   \
   145 288 133 264 260 258 512

Cf. A340801, A006577, A340008, A339991, A340419.

from sympy import isprime
from math import sqrt
def children(N):
    C = []
    if N%2 == 0:
        if isprime(N + 1) == 0: C.append(N+1)
    else:
        p1 = sqrt(N + 2.0); p2 = int(p1 + 0.5)
        if p2**2 == N + 2 and isprime(p2) == 1: C.append(p2)
    C.append(2*N)
    return C
L_last = [1]; print(L_last)
for d in range(1, 18):
    L_1 = []
    for i in range(0, len(L_last)):
        C_i = children(L_last[i])
        for j in range(0, len(C_i)): L_1.append(C_i[j])
    print(L_1); L_last = L_1

A346136 a(n) is the number of iterations that n requires to reach 1 under the map n -> A346063(n).

Original entry on oeis.org

0, 1, 2, 2, 3, 3, 1, 4, 1, 4, 3, 3, 4, 3, 2, 4, 4, 5, 2, 3, 9, 11, 8, 6, 10, 12, 6, 7, 2, 8, 4, 3, 6, 4, 10, 7, 2, 7, 4, 9, 4, 5, 4, 1, 8, 7, 6, 11, 2, 73, 5, 12, 5, 6, 1, 5, 2, 4, 34, 7, 5, 5, 2, 51, 7, 2, 5, 3, 5, 5, 3, 15, 6, 5, 2, 4, 10

Offset: 1

Ya-Ping Lu, Jul 05 2021

Conjecture: the sequence is infinite.

Cf. A339991, A340008, A340418, A340419, A346063.

f(x) = my(k=x^2-1); while(k>3 && !ispseudoprime(k), k\=2); k;
a(n) = my(c=0, x=prime(n)); while(x>2, c++; x=f(x)); c; \\ Jinyuan Wang, Jul 15 2022

from sympy import prime, isprime
for n in range(1, 78):
    m = prime(n); ct = 0
    while m > 2:
        if isprime(m): m = m*m - 1; ct += 1
        else: m //= 2
    print(ct)

a(1) corrected by Jinyuan Wang, Jul 15 2022

cp's OEIS Frontend

A346063 a(n) = primepi(A039634(prime(n)^2-1)).

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A340801 a(n) is the image of n under the map f defined as f(n) = n^2 - 2 if n is an odd prime, f(n) = n/2 if n is even, and f(n) = n - 1 otherwise.

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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.

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A346136 a(n) is the number of iterations that n requires to reach 1 under the map n -> A346063(n).

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