%I #56 Oct 03 2023 13:14:08

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

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

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

%N The number of exponential semiproper divisors of n.

%C An exponential semiproper divisor of n is a divisor d such that rad(d) = rad(n) and gcd(d/rad(n), n/d) = 1, where rad(n) is the largest squarefree divisor of n (A007947).

%C a(n) is also the number of divisors of n that are squares of squarefree numbers (A062503). - _Amiram Eldar_, Oct 08 2022

%C a(n) is also the number of unitary divisors of n that are powerful (A001694). - _Amiram Eldar_, Feb 18 2023

%C The smallest integer that has exactly 2^n exponential semiproper divisors is A061742(n). - _Bernard Schott_, Feb 20 2023

%H Amiram Eldar, <a href="/A323308/b323308.txt">Table of n, a(n) for n = 1..10000</a>

%H Nicusor Minculete, <a href="http://rs.unitbv.ro/BU2013/2014/Series_III/BULETIN%20III%20PDF/4.Minculete-MOD.pdf">A new class of divisors: the exponential semiproper divisors</a>, Bulletin of the Transilvania University of Brasov, Mathematics, Informatics, Physics, Series III, Vol. 7, No. 1 (2014), pp. 37-46.

%H <a href="/index/Eu#epf">Index entries for sequences computed from exponents in factorization of n</a>.

%H <a href="/index/Di#divisors">Index entries for sequences related to divisors of numbers</a>.

%F a(n) = A034444(n/A007947(n)).

%F Multiplicative with a(p^e) = 1 for e = 1 and 2 otherwise.

%F Asymptotic mean: Limit_{n->oo} (1/n) * Sum_{k=1..n} a(k) = 15/Pi^2 = 1.5198177546... (A082020). - _Amiram Eldar_, Nov 08 2020

%F a(n) = Sum_{d^2|n} mu(d)^2. - _Wesley Ivan Hurt_, Feb 13 2022

%F Dirichlet g.f.: zeta(s) * zeta(2*s) / zeta(4*s). - _Werner Schulte_, Dec 29 2022

%F a(n) = A034444(A000188(n)) = A034444(A008833(n)) (the number of unitary divisors of the largest square dividing n). - _Amiram Eldar_, Sep 03 2023

%F a(n) = A034444(A057521(n)) (the number of unitary divisors of the powerful part of n). - _Amiram Eldar_, Oct 03 2023

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

%o (PARI) a(n) = my(f=factor(n)); for (k=1, #f~, f[k,1] = min(f[k,2], 2); f[k,2] = 1); factorback(f); \\ _Michel Marcus_, Jan 11 2019

%Y Cf. A000188, A001694, A007947, A008833, A034444, A057521, A061742, A062503, A082020, A323309, A323310.

%K nonn,easy,mult

%O 1,4

%A _Amiram Eldar_, Jan 10 2019