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A115561 a(n) = lpf((n/lpf(n))/lpf(n/lpf(n))), where lpf=A020639, least prime factor. 6
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,8

COMMENTS

From Peter Munn, Jul 14 2019: (Start)

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)

LINKS

Antti Karttunen, Table of n, a(n) for n = 1..10000

Eric Weisstein's World of Mathematics, Least Prime Factor

FORMULA

a(n) = A020639(A054576(n)).

If A001222(n) >= 3, a(n) = A027746(n,3), otherwise a(n) = 1. - Peter Munn, Jul 13 2019

MATHEMATICA

f[n_] := FactorInteger[n][[1, 1]]; Table[f[#/f@ #] &[n/f@ n], {n, 101}] (* Michael De Vlieger, Aug 14 2017 *)

PROG

(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)))

print map(a, range(1, 102)) # Indranil Ghosh, Aug 12 2017

(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

CROSSREFS

Cf. A001222, A002110, A014673, A020639, A027746, A032742, A054576, A117358, A281889, A281890.

Sequence in context: A077565 A316979 A331024 * A115622 A294892 A108886

Adjacent sequences:  A115558 A115559 A115560 * A115562 A115563 A115564

KEYWORD

nonn

AUTHOR

Reinhard Zumkeller, Mar 10 2006

STATUS

approved

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Last modified July 15 02:31 EDT 2020. Contains 335762 sequences. (Running on oeis4.)