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A368335 The number of divisors of the largest term of A054744 that divides of n.

%I A368335 #11 Apr 26 2025 21:35:56
%S A368335 1,1,1,3,1,1,1,4,1,1,1,3,1,1,1,5,1,1,1,3,1,1,1,4,1,1,4,3,1,1,1,6,1,1,
%T A368335 1,3,1,1,1,4,1,1,1,3,1,1,1,5,1,1,1,3,1,4,1,4,1,1,1,3,1,1,1,7,1,1,1,3,
%U A368335 1,1,1,4,1,1,1,3,1,1,1,5,5,1,1,3,1,1,1
%N A368335 The number of divisors of the largest term of A054744 that divides of n.
%H A368335 Amiram Eldar, <a href="/A368335/b368335.txt">Table of n, a(n) for n = 1..10000</a>
%F A368335 Multiplicative with a(p^e) = 1 if e < p, and a(p^e) = e+1 if e >= p.
%F A368335 a(n) = A000005(A368333(n)).
%F A368335 a(n) >= 1, with equality if and only if n is in A048103.
%F A368335 a(n) <= A000005(n), with equality if and only if n is in A054744.
%F A368335 Dirichlet g.f.: zeta(s)^2 * Product_{p prime} (1 - 1/p^s + 1/p^(p*s-1) + 1/p^((p+1)*s) - 1/p^((p+1)*s-1)).
%F A368335 Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = Product_{p prime} (1 + (1 + (p-1)*p)/((p-1)*p^p)) = 1.98019019497523582894... .
%t A368335 f[p_, e_] := If[e < p, 1, e+1]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100]
%o A368335 (PARI) a(n) = {my(f = factor(n)); prod(i = 1, #f~, if(f[i,2] < f[i,1], 1, f[i,2]+1));}
%Y A368335 Cf. A000005, A048103, A054744, A368331, A368332, A368333, A368334, A368336.
%K A368335 nonn,easy,mult
%O A368335 1,4
%A A368335 _Amiram Eldar_, Dec 21 2023