A366901 The number of exponentially odious divisors of n.

1, 2, 2, 3, 2, 4, 2, 3, 3, 4, 2, 6, 2, 4, 4, 4, 2, 6, 2, 6, 4, 4, 2, 6, 3, 4, 3, 6, 2, 8, 2, 4, 4, 4, 4, 9, 2, 4, 4, 6, 2, 8, 2, 6, 6, 4, 2, 8, 3, 6, 4, 6, 2, 6, 4, 6, 4, 4, 2, 12, 2, 4, 6, 4, 4, 8, 2, 6, 4, 8, 2, 9, 2, 4, 6, 6, 4, 8, 2, 8, 4, 4, 2, 12, 4, 4, 4

Offset: 1

Author

Amiram Eldar, Oct 27 2023

Keywords

• nonn
• easy
• mult

Comments

First differs from A049599 and A282446 at n = 32, from A365551 at n = 64, and from A353898 at n = 128.

The number of divisors of n that are exponentially odious numbers (A270428), i.e., numbers having only odious (A000069) exponents in their canonical prime factorization.

The sum of these divisors is A366903(n) and the largest of them is A366905(n).

Links

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

Crossrefs

Cf. A000005, A000069, A004709, A115384, A270428, A366903, A366905.

Cf. A049599, A282446, A365551, A353898.

Similar sequences: A325837, A353898, A365680, A366902.

Programs

Mathematica

f[p_, e_] := Floor[e/2] + If[OddQ[e] || EvenQ[DigitCount[e + 1, 2, 1]], 1, 0] + 1; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100]

PARI

s(n) = 1 + n\2 + (n%2 || hammingweight(n+1)%2==0); \\ after Charles R Greathouse IV at A115384
a(n) = vecprod(apply(x -> s(x), factor(n)[, 2]));

Formula

Multiplicative with a(p^e) = A115384(e) + 1.

a(n) <= A000005(n), with equality if and only if n is a cubefree number (A004709).