A086320 a(n) is the depth of the prime tree formed when 4p +- 3 is applied to the n-th prime and repeatedly to any primes generated from the n-th prime via this process.

11, 1, 6, 3, 10, 1, 3, 6, 5, 3, 2, 3, 2, 1, 9, 1, 6, 3, 3, 2, 1, 5, 1, 4, 1, 3, 2, 3, 4, 2, 1, 3, 1, 1, 3, 2, 3, 1, 1, 1, 5, 2, 8, 3, 1, 1, 1, 1, 2, 3, 5, 2, 2, 1, 3, 2, 1, 2, 1, 1, 4, 1, 2, 1, 4, 1, 5, 1, 1, 2, 3, 2, 3, 3, 1, 1, 2, 1, 2, 1, 1, 1, 2, 1, 3, 1, 1, 4, 2, 1, 5, 4, 2, 1, 3, 1, 2, 2, 6, 4, 1, 1, 1, 2

Offset

1,1

Comments

Note all prime trees have a minimum depth of 1, as the starting prime forms the root of the tree.

Links

Alois P. Heinz, Table of n, a(n) for n = 1..65536

Example

a(125) = 5 because the 125th prime is 691, which generates further primes through 4 repeated applications of 4p +- 3, giving a prime tree with generations as follows:
1. 691
2. 4 * 691 + 3 = 2767
3. 4 * 2767 + 3 = 11071
4. 4 * 11071 - 3 = 44281
5. 4 * 44281 + 3 = 177127

Maple

b:= proc(p) option remember;
      `if`(isprime(p), 1 + max(b(4*p+3), b(4*p-3)), 0)
    end:
a:= n-> b(ithprime(n)):
seq(a(n), n=1..120);  # Alois P. Heinz, Dec 02 2018

Mathematica

f[n_] := f[n] = If[PrimeQ[n], 1 + Max[f[4 n - 3], f[4 n + 3]], 0]; f /@ Prime@Range@100 (* Amiram Eldar, Dec 02 2018 *)

Prog

(Python)
from functools import cache
from sympy import isprime, prime
@cache
def b(p): return 1 + max(b(4*p+3), b(4*p-3)) if isprime(p) else 0
def a(n): return b(prime(n))
print([a(n) for n in range(1, 105)]) # Michael S. Branicky, Nov 01 2021 after Alois P. Heinz

Crossrefs

Cf. A086319.

Sequence in context: A110305 A010199 A010200 * A185540 A095193 A374332

Adjacent sequences: A086317 A086318 A086319 * A086321 A086322 A086323

Keyword

nonn

Author

Chuck Seggelin (barkeep(AT)plastereddragon.com), Jul 17 2003

Extensions

Offset corrected by Alois P. Heinz, Dec 02 2018

Status

approved