A368979 The number of exponential 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

Offset: 1

Author

Amiram Eldar, Jan 11 2024

Keywords

• nonn
• easy
• mult

Comments

First differs from A367516 at n = 128, and from A359411 at n = 512.

Links

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

Crossrefs

Cf. A001227, A008578, A049419, A138302, A268335, A322791, A368980.

Cf. A359411, A367516.

Similar sequences: A055076, A322483, A363825, A368977.

Programs

Mathematica

f[p_, e_] := DivisorSigma[0, e/2^IntegerExponent[e, 2]]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100]

PARI

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

Formula

Multiplicative with a(p^e) = A001227(e).

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

a(n) <= A049419(n), with equality if and only if n is noncomposite (A008578).

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) = 1.13657098749361390865..., where d(k) is the number of odd divisors of k (A001227).