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A368334 The number of terms of A054744 that are unitary divisors of n.

%I A368334 #10 Apr 26 2025 21:35:49
%S A368334 1,1,1,2,1,1,1,2,1,1,1,2,1,1,1,2,1,1,1,2,1,1,1,2,1,1,2,2,1,1,1,2,1,1,
%T A368334 1,2,1,1,1,2,1,1,1,2,1,1,1,2,1,1,1,2,1,2,1,2,1,1,1,2,1,1,1,2,1,1,1,2,
%U A368334 1,1,1,2,1,1,1,2,1,1,1,2,2,1,1,2,1,1,1
%N A368334 The number of terms of A054744 that are unitary divisors of n.
%C A368334 First differs from A081117 at n = 28.
%C A368334 Also, the number of terms of A072873 that are unitary divisors of n.
%H A368334 Amiram Eldar, <a href="/A368334/b368334.txt">Table of n, a(n) for n = 1..10000</a>
%F A368334 Multiplicative with a(p^e) = 1 if e < p, and a(p^e) = 2 if e >= p.
%F A368334 a(n) = A034444(A368333(n)).
%F A368334 a(n) = A034444(A327939(n)).
%F A368334 a(n) >= 1, with equality if and only if n is in A048103.
%F A368334 a(n) <= A034444(n), with equality if and only if n is in A054744.
%F A368334 Dirichlet g.f.: zeta(s) * Product_{p prime} (1 + 1/p^(p*s)).
%F A368334 Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = Product_{p prime} (1 + 1/p^p) = 1.29671268566745796443... .
%t A368334 f[p_, e_] := If[e < p, 1, 2]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100]
%o A368334 (PARI) a(n) = {my(f = factor(n)); prod(i = 1, #f~, if(f[i,2] < f[i,1], 1, 2));}
%Y A368334 Cf. A034444, A048103, A054744, A072873, A327939, A368330, A368332, A368333, A368335.
%Y A368334 Cf. A081117.
%K A368334 nonn,easy,mult
%O A368334 1,4
%A A368334 _Amiram Eldar_, Dec 21 2023