A115561 a(n) = lpf((n/lpf(n))/lpf(n/lpf(n))), where lpf=A020639, least prime factor.

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

Offset: 1

Author

Reinhard Zumkeller, Mar 10 2006

Keywords

• nonn

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

Crossrefs

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

Programs

Mathematica

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

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([a(n) for n in range(1, 102)]) # Indranil Ghosh, Aug 12 2017

Formula

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

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