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%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
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