A323308 The number of exponential semiproper divisors of n.

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, 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, 1, 1, 1, 4, 1, 1, 2, 2, 1, 1, 1, 2, 2, 1, 1, 2, 1, 1, 1

Offset: 1

Author

Amiram Eldar, Jan 10 2019

Keywords

• nonn
• easy
• mult

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.
• Index entries for sequences computed from exponents in factorization of n.
• Index entries for sequences related to divisors of numbers.

Crossrefs

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

Programs

Mathematica

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

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

Formula

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

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