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

1,8

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

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).

Mathematica

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

Prog

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

Crossrefs

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

Cf. A359411, A367516.

Similar sequences: A055076, A322483, A363825, A368977.

Sequence in context: A368168 A359411 A367516 * A325989 A055229 A367698

Adjacent sequences: A368976 A368977 A368978 * A368980 A368981 A368982

Keyword

nonn,easy,mult

Author

Amiram Eldar, Jan 11 2024

Status

approved