A368330 - OEIS

%I #6 Dec 21 2023 21:15:15

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

%N The number of terms of A054743 that are unitary divisors of n.

%C First differ from A043281 at n = 49.

%H Amiram Eldar, <a href="/A368330/b368330.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(A368329(n)).	

%F a(n) >= 1, with equality if and only if n is in A207481.

%F a(n) <= A034444(n), with equality if and only if n is in A054743.

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

%F Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = Product_{p prime} (1 + 1/p^(p+1)) = 1.13896197534988330925... .

%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, A054743, A207481, A368328, A368329, A368331, A368334.

%Y Cf. A043281.

%K nonn,easy,mult

%O 1,8

%A _Amiram Eldar_, Dec 21 2023