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%I #15 Jan 15 2024 01:47:50
%S 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,
%T 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,
%U 2,2,2,3,2,2,3,3,2,2,2,5,5,2,2,3,2,2,2,2,2,3
%N Largest prime factor of number of divisors of n; a(1)=1.
%C From _Robert Israel_, Dec 04 2016: (Start)
%C a(n)=2 if and only if every member of the prime signature of n is of the form 2^k-1.
%C a(m*k) = max(a(m),a(k)) if m and k are coprime. (End)
%H Reinhard Zumkeller, <a href="/A071188/b071188.txt">Table of n, a(n) for n = 1..10000</a>
%F a(n) = A006530(A000005(n)).
%F 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
%p f:= n -> max(1, numtheory:-factorset(numtheory:-tau(n))):
%p map(f, [$1..100]); # _Robert Israel_, Dec 04 2016
%t Max[Transpose[FactorInteger[#]][[1]]]&/@DivisorSigma[0,Range[100]] (* _Harvey P. Dale_, Aug 28 2013 *)
%o (Haskell)
%o a071188 = a006530 . a000005 -- _Reinhard Zumkeller_, Sep 04 2013
%o (PARI) a(n) = if(n == 1, 1, vecmax(factor(numdiv(n))[, 1])); \\ _Michel Marcus_, Dec 05 2016
%Y Cf. A000005, A006530, A071187, A071190, A078542, A078543, A078544, A124010.
%Y Cf. A327839, A354181.
%K nonn,easy
%O 1,2
%A _Reinhard Zumkeller_, May 15 2002
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