cp's OEIS Frontend

The function S_e(n) in Sándor (2006).

A number k is an exponential harmonic of type 2 (A348964) if and only if a(k) | k * A049419(k).

Equivalently, the asymptotic density of numbers with an odd number of exponential divisors (A049419).

First differs from A073184 at n = 64.

First differs from A383865 at n = 256.

The number of divisors d of n such that each is a unitary divisor of an exponential unitary divisor of n (see A361255).

Analogous to the number of (1+e)-divisors (A049599) as exponential unitary divisors (A361255, A278908) are analogous to exponential divisors (A322791, A049419).

The sum of these divisors is A383864(n).

Also, the number of divisors d of n having the property that for every prime p dividing n the p-adic valuation of d is either 0 or a squarefree divisor of the p-adic valuation of n. The sum of these divisors is A383867(n).

Includes all the odd squarefree numbers (A056911). First differs from this sequence at n = 34.

Equivalently, the odd terms of A197680, i.e., odd numbers with an odd number of exponential divisors (A049419).

The asymptotic density of this sequence is 0.409797... (A357017).

First differs from A377019 at n = 55: A377019(55) = 64 is not a term of this sequence.

First differs from A344742 at n = 62: A344742(62) = 72 is not a term of this sequence.

All the cubefree numbers (A004709) are terms. The least term that is not cubefree is a(215) = 256 = 2^8. The cubefree numbers are the terms whose number of coreful divisors is equal to their number of exponential divisors.

All the exponentially refactorable numbers (A382065) are terms. The least term that is not in A382065 is a(362) = 432 = 2^4 * 3^3. The next terms that are not in A382065 are 648, 2000, 2160, 3024, 3240, 4536, 4752, 5000, ... .

For a number k whose prime factorization is Product_{i} p_i^e_i, a coreful divisor d of k has the prime factorization Product_{i} p_i^f_i with f_i >= 1 for all i. An exponential divisor of k is a coreful divisor with the additional condition that f_i | e_i for all i.

Numbers k such that A049419(k) | A005361(k).

The criterion according to which a number belongs to this sequence depends only on the prime signature of this number: if {e_1, e_2, ... } are the exponents in the prime factorization of k then k is a term if and only if A005361(k)/A049419(k) = Product_{i} e_i/A000005(e_i) is an integer.

A number k is a term if and only if the cubefull part of k, A360540(k), is a term. Therefore, the primitive terms of this sequence are the cubefull terms, A382064.

The asymptotic density of this sequence is Sum_{n>=1} f(A382064(n)) = 0.83697905945047..., where f(n) = (1/(zeta(3)*n)) * Product_{prime p|n} (p^2/(p^2+p+1)).

First differs from A383863 at n = 256.

The number of divisors d of n such that each is a unitary divisor of an exponential infinitary divisor of n (see A383760).

Analogous to the number of (1+e)-divisors (A049599) as exponential infinitary divisors (A383760, A307848) are analogous to exponential divisors (A322791, A049419).

The sum of these divisors is A383866(n).

The exponential divisors of a number n = Product p(i)^e(i) are all numbers of the form Product p(i)^s(i) where s(i) divides e(i) for all i.

Not multiplicative: a(3)=3 (e-divisor 3^1), a(4)=8 (e-divisors 2^1 and 2^2), but a(12)=72 (e-divisors 3*2 and 3*2^2) <> a(3)*a(4). - R. J. Mathar, Apr 14 2011

The non-exponential divisors d|n of a number n = Product_i p(i)^e(i) are divisors d not of the form Product_i p(i)^s(i), s(i)|e(i) for all i.

A divisor d of k is tri-unitary if the greatest common bi-unitary divisor of d and k/d is 1.

Differs from A037445 at n = 32, 96, 128, 160, 224, ...

Sándor (2006) proved that all the squarefree numbers are exponential harmonic numbers of type 1.