A384557 The number of exponential unitary (or e-unitary) divisors of n that are exponentially odd numbers (A268335).

1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 2, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 2, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1

Offset: 1

Author

Amiram Eldar, Jun 03 2025

Keywords

• nonn
• easy
• mult

Comments

First differs from A359411 at n = 2097152 = 2^21: a(2097152) = 4, while A359411(2097152) = 2.

First differs from A368979 at n = 512 = 2^9: a(512) = 2, while A368979(512) = 3.

First differs from A367516 at n = 128 = 2^7: a(128) = 2, while A367516(128) = 1.

First differs from A382291 at n = 128 = 2^7: a(128) = 2, while A382291(128) = 4.

First differs from A368168 at n = 64 = 2^6: a(64) = 2, while A368168(64) = 1.

The sum of these divisors is A384559(n), and the largest of them is A331737(n).

The number of exponential unitary (or e-unitary) divisors of n is A278908(n) and the number of divisors of n that are exponentially odd numbers is A322483(n).

All the terms are powers of 2. The first term that is greater than 2 is a(32768) = 4.

Links

• Amiram Eldar, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A068068, A138302, A268335, A278908, A322483, A331737, A359411, A367516, A368168, A368979, A382291, A384559.

Programs

Mathematica

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

PARI

a(n) = vecprod(apply(x -> 2^omega(x >> valuation(x, 2)) , factor(n)[, 2]));

Formula

Multiplicative with a(p^e) = A068068(e).

a(n) >= 1, with equality if and only if n is in A138302.

a(n) <= A278908(n), with equality if and only if n is an exponentially odd number.

Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = Product_{p prime} (1 + Sum_{k>=2} (d(k) - d(k-1))/p^k) = 1.13551542615965557947..., where d(k) is the number of odd unitary divisors of k (A068068).