A391988 - OEIS

%I #9 Dec 29 2025 04:02:44

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

%T 4,1,1,1,2,1,1,1,2,2,1,1,2,2,1,2,1,2,1,2,1,1,1,2,1,1,2,4,1,1,1,2,1,1,

%U 1,4,1,1,2,2,1,1,1,1,1,2,1,1,1,2,1,2,1,2

%N The number of exponential divisors of the exponentially squarefree numbers.

%C First differs from A388973 at n = 246.

%C Also, the number of exponential unitary divisors of the exponentially squarefree numbers.

%H Amiram Eldar, <a href="/A391988/b391988.txt">Table of n, a(n) for n = 1..10000</a>

%F a(n) = A049419(A209061(n)).

%F a(n) = A278908(A209061(n)).

%F Asymptotic mean: lim_{m->oo} (1/m) * Sum_{k=1..m} a(k) = (1/d) * Product_{p prime} (1 + Sum_{k>=2} (A074823(k) - A074823(k-1))/p^k) = 1.49848918426328501463..., where d = A262276.

%t s[n_] := Module[{e = FactorInteger[n][[;;, 2]]}, Boole[AllTrue[e, SquareFreeQ]] * Times @@ (2^PrimeNu[e])]; s[1] = 1; Select[Array[s, 100], # > 0 &]

%o (PARI) f(n) = {my(e = factor(n)[, 2]); for(i = 1, #e, if(!issquarefree(e[i]), return(0))); vecprod(apply(x -> 2^omega(x), e));}

%o list(lim) = select(x -> x > 0, vector(lim, i, f(i)));

%Y Cf. A049419, A074823, A209061, A262276, A278908, A388973, A391989.

%K nonn,easy

%O 1,4

%A _Amiram Eldar_, Dec 26 2025