A086320 - OEIS

%I #26 Nov 01 2021 03:21:40

%S 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,

%T 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,

%U 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

%N 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.

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

%H Alois P. Heinz, <a href="/A086320/b086320.txt">Table of n, a(n) for n = 1..65536</a>

%e 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:

%e 1. 691

%e 2. 4 * 691 + 3 = 2767

%e 3. 4 * 2767 + 3 = 11071

%e 4. 4 * 11071 - 3 = 44281

%e 5. 4 * 44281 + 3 = 177127

%p b:= proc(p) option remember;

%p       `if`(isprime(p), 1 + max(b(4*p+3), b(4*p-3)), 0)

%p     end:

%p a:= n-> b(ithprime(n)):

%p seq(a(n), n=1..120);  # _Alois P. Heinz_, Dec 02 2018

%t 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 *)

%o (Python)

%o from functools import cache

%o from sympy import isprime, prime

%o @cache

%o def b(p): return 1 + max(b(4*p+3), b(4*p-3)) if isprime(p) else 0

%o def a(n): return b(prime(n))

%o print([a(n) for n in range(1, 105)]) # _Michael S. Branicky_, Nov 01 2021 after _Alois P. Heinz_

%Y Cf. A086319.

%K nonn

%O 1,1

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

%E Offset corrected by _Alois P. Heinz_, Dec 02 2018