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

Author

Ya-Ping Lu, Jul 03 2021

Keywords

• nonn

Comments

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.

Links

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

Crossrefs

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

Programs

Mathematica

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

Python

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))

Formula

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