A368334 - OEIS

%I #7 Dec 21 2023 21:15:48

%S 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 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 1,1,1,2,1,1,1,2,1,1,1,2,2,1,1,2,1,1,1

%N The number of terms of A054744 that are unitary divisors of n.

%C First differs from A081117 at n = 28.

%C Also, the number of terms of A072873 that are unitary divisors of n.

%H Amiram Eldar, <a href="/A368334/b368334.txt">Table of n, a(n) for n = 1..10000</a>

%F Multiplicative with a(p^e) = 1 if e < p, and a(p^e) = 2 if e >= p.

%F a(n) = A034444(A368333(n)).	

%F a(n) = A034444(A327939(n)).

%F a(n) >= 1, with equality if and only if n is in A048103.

%F a(n) <= A034444(n), with equality if and only if n is in A054744.

%F Dirichlet g.f.: zeta(s) * Product_{p prime} (1 + 1/p^(p*s)).

%F Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = Product_{p prime} (1 + 1/p^p) = 1.29671268566745796443... .

%t f[p_, e_] := If[e < p, 1, 2]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100]

%o (PARI) a(n) = {my(f = factor(n)); prod(i = 1, #f~, if(f[i,2] < f[i,1], 1, 2));}

%Y Cf. A034444, A048103, A054744, A072873, A327939, A368330, A368332, A368333, A368335.

%Y Cf. A081117.

%K nonn,easy,mult

%O 1,4

%A _Amiram Eldar_, Dec 21 2023