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A368330 The number of terms of A054743 that are unitary divisors of n.

%I A368330 #9 Apr 26 2025 21:35:29
%S A368330 1,1,1,1,1,1,1,2,1,1,1,1,1,1,1,2,1,1,1,1,1,1,1,2,1,1,1,1,1,1,1,2,1,1,
%T A368330 1,1,1,1,1,2,1,1,1,1,1,1,1,2,1,1,1,1,1,1,1,2,1,1,1,1,1,1,1,2,1,1,1,1,
%U A368330 1,1,1,2,1,1,1,1,1,1,1,2,2,1,1,1,1,1,1
%N A368330 The number of terms of A054743 that are unitary divisors of n.
%C A368330 First differ from A043281 at n = 49.
%H A368330 Amiram Eldar, <a href="/A368330/b368330.txt">Table of n, a(n) for n = 1..10000</a>
%F A368330 Multiplicative with a(p^e) = 1 if e <= p, and a(p^e) = 2 if e > p.
%F A368330 a(n) = A034444(A368329(n)).
%F A368330 a(n) >= 1, with equality if and only if n is in A207481.
%F A368330 a(n) <= A034444(n), with equality if and only if n is in A054743.
%F A368330 Dirichlet g.f.: zeta(s) * Product_{p prime} (1 + 1/p^((p+1)*s)).
%F A368330 Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = Product_{p prime} (1 + 1/p^(p+1)) = 1.13896197534988330925... .
%t A368330 f[p_, e_] := If[e <= p, 1, 2]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100]
%o A368330 (PARI) a(n) = {my(f = factor(n)); prod(i = 1, #f~, if(f[i,2] <= f[i,1], 1, 2));}
%Y A368330 Cf. A034444, A054743, A207481, A368328, A368329, A368331, A368334.
%Y A368330 Cf. A043281.
%K A368330 nonn,easy,mult
%O A368330 1,8
%A A368330 _Amiram Eldar_, Dec 21 2023