Year-end appeal: Please make a donation to the OEIS Foundation to support ongoing development and maintenance of the OEIS. We are now in our 61st year, we have over 378,000 sequences, and we’ve reached 11,000 citations (which often say “discovered thanks to the OEIS”).
%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