A363825 The number of infinitary divisors of n that are exponentially odd numbers (A268335).

1, 2, 2, 1, 2, 4, 2, 3, 1, 4, 2, 2, 2, 4, 4, 1, 2, 2, 2, 2, 4, 4, 2, 6, 1, 4, 3, 2, 2, 8, 2, 3, 4, 4, 4, 1, 2, 4, 4, 6, 2, 8, 2, 2, 2, 4, 2, 2, 1, 2, 4, 2, 2, 6, 4, 6, 4, 4, 2, 4, 2, 4, 2, 1, 4, 8, 2, 2, 4, 8, 2, 3, 2, 4, 2, 2, 4, 8, 2, 2, 1, 4, 2, 4, 4, 4, 4

Offset: 1

Author

Amiram Eldar, Oct 19 2023

Keywords

• nonn
• easy
• mult

Comments

First differs from A295878 at n = 32.

Links

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

Crossrefs

Cf. A000005, A000120, A005117, A037445, A077609, A268335, A295878, A358260.

Programs

Mathematica

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

PARI

a(n) = {my(f = factor(n)); prod(i = 1, #f~, if(f[i, 2]%2, 2^hammingweight(f[i, 2]-1) + 1, 1))};

Formula

Multiplicative with a(p^e) = 1 if e is even, and a(p^e) = 2^A000120(e-1) + 1 if e is odd.

a(n) <= A000005(n) with equality if and only if n is squarefree (A005117).