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A380086 The number of unitary divisors of n that are terms in A276078.

%I A380086 #11 Jan 12 2025 03:55:01
%S A380086 1,2,2,1,2,4,2,1,2,4,2,2,2,4,4,1,2,4,2,2,4,4,2,2,2,4,1,2,2,8,2,1,4,4,
%T A380086 4,2,2,4,4,2,2,8,2,2,4,4,2,2,2,4,4,2,2,2,4,2,4,4,2,4,2,4,4,1,4,8,2,2,
%U A380086 4,8,2,2,2,4,4,2,4,8,2,2,1,4,2,4,4,4,4
%N A380086 The number of unitary divisors of n that are terms in A276078.
%H A380086 Amiram Eldar, <a href="/A380086/b380086.txt">Table of n, a(n) for n = 1..10000</a>
%F A380086 a(n) = A034444(A380085(n)).
%F A380086 Multiplicative with a(p^e) = 2 if e <= pi(p) = A000720(p), and 1 otherwise.
%F A380086 a(n) = 1 if and only if n is in A325127.
%F A380086 a(n) < A034444(n) if and only if n is in A276079.
%F A380086 a(n) = A034444(n) if and only if n is in A276078.
%F A380086 a(n) = A377516(n) if and only if n is squarefree (A005117).
%F A380086 Dirichlet g.f.: zeta(s) * Product_{p prime} (1 + 1/p^s - 1/p^((pi(p)+1)*s)).
%F A380086 Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k)/A034444(k) = Product_{p prime} (1 - 1/(2*p^(pi(p)+1))) = 0.85808348184674088116... .
%t A380086 f[p_, e_] := If[e <= PrimePi[p], 2, 1]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100]
%o A380086 (PARI) a(n) = {my(f = factor(n)); prod(i = 1, #f~,if(f[i,2] <= primepi(f[i,1]), 2, 1));}
%Y A380086 Cf. A000720, A005117, A034444, A077610, A276078, A325127, A377516, A380085, A380087, A380089.
%K A380086 nonn,easy,mult
%O A380086 1,2
%A A380086 _Amiram Eldar_, Jan 11 2025