OFFSET
1,4
COMMENTS
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).
a(n) is also the number of divisors of n that are squares of squarefree numbers (A062503). - Amiram Eldar, Oct 08 2022
a(n) is also the number of unitary divisors of n that are powerful (A001694). - Amiram Eldar, Feb 18 2023
The smallest integer that has exactly 2^n exponential semiproper divisors is A061742(n). - Bernard Schott, Feb 20 2023
LINKS
Amiram Eldar, Table of n, a(n) for n = 1..10000
Nicusor Minculete, A new class of divisors: the exponential semiproper divisors, Bulletin of the Transilvania University of Brasov, Mathematics, Informatics, Physics, Series III, Vol. 7, No. 1 (2014), pp. 37-46.
FORMULA
Multiplicative with a(p^e) = 1 for e = 1 and 2 otherwise.
Asymptotic mean: Limit_{n->oo} (1/n) * Sum_{k=1..n} a(k) = 15/Pi^2 = 1.5198177546... (A082020). - Amiram Eldar, Nov 08 2020
a(n) = Sum_{d^2|n} mu(d)^2. - Wesley Ivan Hurt, Feb 13 2022
Dirichlet g.f.: zeta(s) * zeta(2*s) / zeta(4*s). - Werner Schulte, Dec 29 2022
a(n) = A034444(A000188(n)) = A034444(A008833(n)) (the number of unitary divisors of the largest square dividing n). - Amiram Eldar, Sep 03 2023
a(n) = A034444(A057521(n)) (the number of unitary divisors of the powerful part of n). - Amiram Eldar, Oct 03 2023
MATHEMATICA
f[p_, e_] := If[e==1, 1, 2]; a[1]=1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100]
PROG
(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
CROSSREFS
KEYWORD
nonn,easy,mult
AUTHOR
Amiram Eldar, Jan 10 2019
STATUS
approved