A071188 Largest prime factor of number of divisors of n; a(1)=1.

1, 2, 2, 3, 2, 2, 2, 2, 3, 2, 2, 3, 2, 2, 2, 5, 2, 3, 2, 3, 2, 2, 2, 2, 3, 2, 2, 3, 2, 2, 2, 3, 2, 2, 2, 3, 2, 2, 2, 2, 2, 2, 2, 3, 3, 2, 2, 5, 3, 3, 2, 3, 2, 2, 2, 2, 2, 2, 2, 3, 2, 2, 3, 7, 2, 2, 2, 3, 2, 2, 2, 3, 2, 2, 3, 3, 2, 2, 2, 5, 5, 2, 2, 3, 2, 2, 2, 2, 2, 3

Offset: 1

Author

Reinhard Zumkeller, May 15 2002

Keywords

• nonn
• easy

Comments

From Robert Israel, Dec 04 2016: (Start)

a(n)=2 if and only if every member of the prime signature of n is of the form 2^k-1.

a(m*k) = max(a(m),a(k)) if m and k are coprime. (End)

Links

• Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A000005, A006530, A071187, A071190, A078542, A078543, A078544, A124010.

Cf. A327839, A354181.

Programs

Haskell

a071188 = a006530 . a000005  -- Reinhard Zumkeller, Sep 04 2013

Maple

f:= n -> max(1, numtheory:-factorset(numtheory:-tau(n))):
map(f, [$1..100]); # Robert Israel, Dec 04 2016

Mathematica

Max[Transpose[FactorInteger[#]][[1]]]&/@DivisorSigma[0,Range[100]] (* Harvey P. Dale, Aug 28 2013 *)

PARI

a(n) = if(n == 1, 1, vecmax(factor(numdiv(n))[, 1])); \\ Michel Marcus, Dec 05 2016

Formula

a(n) = A006530(A000005(n)).

Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = 2*d(1) + Sum_{k>=2} prime(k)*(d(k) - d(k-1)) = 2.4365518864..., where d(1) = A327839, and for k >= 2, d(k) is the asymptotic density of numbers whose number of divisors is a prime(k)-smooth number, i.e., d(k) = Product_{p prime} ((1 - 1/p) * Sum_{i, A006530(i) <= prime(k)} 1/p^(i-1)) (see A354181 for an example). - Amiram Eldar, Jan 15 2024