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A115561
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a(n) = lpf((n/lpf(n))/lpf(n/lpf(n))), where lpf=A020639, least prime factor.
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6
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1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 3, 1, 1, 1, 2, 1, 3, 1, 5, 1, 1, 1, 2, 1, 1, 3, 7, 1, 5, 1, 2, 1, 1, 1, 3, 1, 1, 1, 2, 1, 7, 1, 11, 5, 1, 1, 2, 1, 5, 1, 13, 1, 3, 1, 2, 1, 1, 1, 3, 1, 1, 7, 2, 1, 11, 1, 17, 1, 7, 1, 2, 1, 1, 5, 19, 1, 13, 1, 2, 3, 1, 1, 3, 1, 1, 1, 2, 1, 3, 1, 23, 1, 1, 1, 2, 1, 7, 11, 5, 1
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OFFSET
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1,8
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COMMENTS
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a(n) = 1 if and only if n is 1 or a prime or semiprime. Otherwise a(n) is the 3rd factor when n is written as a product of primes in nondecreasing order. For example, 60 = 2*2*3*5, so a(60) = 3.
Although values equal to 1 are predominant at low indices, their asymptotic density is 0, whereas for values equal to prime(k) for k > 0 the asymptotic density is positive, namely A281890(k,3)/A002110(k)^3. For all sufficiently large n the median value of a(1), a(2), ... a(n) is A281889(3) = 433.
(End)
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LINKS
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FORMULA
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MATHEMATICA
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f[n_] := FactorInteger[n][[1, 1]]; Table[f[#/f@ #] &[n/f@ n], {n, 101}] (* Michael De Vlieger, Aug 14 2017 *)
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PROG
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(Python)
from sympy import divisors, primefactors
def a032752(n): return 1 if n==1 else divisors(n)[-2]
def a020639(n): return 1 if n==1 else primefactors(n)[0]
def a(n): return a020639(a032752(a032752(n)))
(PARI) a020639(n) = if(n>1, if(n>n=factor(n, 0)[1, 1], n, factor(n)[1, 1]), 1) \\ after M. F. Hasler in A020639
a(n) = a020639((n/a020639(n))/a020639(n/a020639(n))) \\ Felix Fröhlich, Jul 15 2019
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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