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A368332 The number of terms of A054744 that divide n.

%I A368332 #6 Dec 21 2023 21:15:31
%S A368332 1,1,1,2,1,1,1,3,1,1,1,2,1,1,1,4,1,1,1,2,1,1,1,3,1,1,2,2,1,1,1,5,1,1,
%T A368332 1,2,1,1,1,3,1,1,1,2,1,1,1,4,1,1,1,2,1,2,1,3,1,1,1,2,1,1,1,6,1,1,1,2,
%U A368332 1,1,1,3,1,1,1,2,1,1,1,4,3,1,1,2,1,1,1
%N A368332 The number of terms of A054744 that divide n.
%C A368332 The number of divisors d of n such that e >= p for all prime powers p^e in the prime factorization of d (i.e., e >= 1 and p^(e+1) does not divide d).
%C A368332 The largest of these divisors is A368333(n).
%H A368332 Amiram Eldar, <a href="/A368332/b368332.txt">Table of n, a(n) for n = 1..10000</a>
%F A368332 Multiplicative with a(p^e) = 1 if e < p, and a(p^e) = e - p + 2 if e >= p.
%F A368332 a(n) >= 1, with equality if and only if n is in A048103.
%F A368332 Dirichlet g.f.: zeta(s)^2 * Product_{p prime} (1 - 1/p^s + 1/p^(p*s)).
%F A368332 Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = Product_{p prime} (1 + 1/((p-1)*p^(p-1))) = 1.58396891058853238595... .
%t A368332 f[p_, e_] := If[e < p, 1, e - p + 2]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100]
%o A368332 (PARI) a(n) = {my(f = factor(n)); prod(i = 1, #f~, if(f[i,2] < f[i,1], 1, f[i,2] - f[i,1] + 2));}
%Y A368332 Cf. A054744, A365632, A368328, A368333, A368334, A368335.
%K A368332 nonn,easy,mult
%O A368332 1,4
%A A368332 _Amiram Eldar_, Dec 21 2023