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A322028
Number of distinct orders of primeness among the prime factors of n.
3
0, 1, 1, 1, 1, 2, 1, 1, 1, 2, 1, 2, 1, 1, 2, 1, 1, 2, 1, 2, 2, 2, 1, 2, 1, 1, 1, 1, 1, 3, 1, 1, 2, 2, 2, 2, 1, 1, 2, 2, 1, 2, 1, 2, 2, 1, 1, 2, 1, 2, 1, 1, 1, 2, 2, 1, 2, 1, 1, 3, 1, 2, 2, 1, 2, 3, 1, 2, 2, 2, 1, 2, 1, 1, 2, 1, 2, 2, 1, 2, 1, 2, 1, 2, 2, 1, 2
OFFSET
1,6
COMMENTS
The order of primeness (A078442) of a prime number p is the number of times one must apply A000720 to obtain a nonprime number.
LINKS
N. Fernandez, An order of primeness [cached copy, included at A006450 with permission of the author]
EXAMPLE
a(105) = 3 because the prime factors of 105 = 3*5*7 have 3 different orders of primeness, namely 2, 3, and 1 respectively.
MAPLE
with(numtheory):
p:= proc(n) option remember;
`if`(isprime(n), 1+p(pi(n)), 0)
end:
a:= n-> nops(map(p, factorset(n))):
seq(a(n), n=1..120); # Alois P. Heinz, Nov 24 2018
MATHEMATICA
Table[If[n==1, 0, Length[Union[Length[NestWhileList[PrimePi, PrimePi[#], PrimeQ]]&/@FactorInteger[n][[All, 1]]]]], {n, 100}]
KEYWORD
nonn
AUTHOR
Gus Wiseman, Nov 24 2018
STATUS
approved