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A368328 The number of terms of A054743 that divide n.

%I A368328 #8 Dec 21 2023 21:14:56
%S A368328 1,1,1,1,1,1,1,2,1,1,1,1,1,1,1,3,1,1,1,1,1,1,1,2,1,1,1,1,1,1,1,4,1,1,
%T A368328 1,1,1,1,1,2,1,1,1,1,1,1,1,3,1,1,1,1,1,1,1,2,1,1,1,1,1,1,1,5,1,1,1,1,
%U A368328 1,1,1,2,1,1,1,1,1,1,1,3,2,1,1,1,1,1,1
%N A368328 The number of terms of A054743 that divide n.
%C A368328 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 A368328 The largest of these divisors is A368329(n).
%H A368328 Amiram Eldar, <a href="/A368328/b368328.txt">Table of n, a(n) for n = 1..10000</a>
%F A368328 Multiplicative with a(p^e) = 1 if e <= p, and a(p^e) = e - p + 1 if e > p.
%F A368328 a(n) >= 1, with equality if and only if n is in A207481.
%F A368328 Dirichlet g.f.: zeta(s)^2 * Product_{p prime} (1 - 1/p^s + 1/p^((p+1)*s)).
%F A368328 Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = Product_{p prime} (1 + 1/((p-1)*p^p)) = 1.27325025767774256043... .
%t A368328 f[p_, e_] := If[e <= p, 1, e - p + 1]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100]
%o A368328 (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] + 1));}
%Y A368328 Cf. A054743, A207481, A365632, A368329, A368330, A368331, A368332.
%K A368328 nonn,easy,mult
%O A368328 1,8
%A A368328 _Amiram Eldar_, Dec 21 2023