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A086320
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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.
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1
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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
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OFFSET
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1,1
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COMMENTS
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Note all prime trees have a minimum depth of 1, as the starting prime forms the root of the tree.
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LINKS
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EXAMPLE
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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
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MAPLE
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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)):
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MATHEMATICA
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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 *)
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PROG
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(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))
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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Chuck Seggelin (barkeep(AT)plastereddragon.com), Jul 17 2003
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EXTENSIONS
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STATUS
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approved
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