A368172 The number of divisors of the largest cubefull exponentially odd divisor of n (A368170).

1, 1, 1, 1, 1, 1, 1, 4, 1, 1, 1, 1, 1, 1, 1, 4, 1, 1, 1, 1, 1, 1, 1, 4, 1, 1, 4, 1, 1, 1, 1, 6, 1, 1, 1, 1, 1, 1, 1, 4, 1, 1, 1, 1, 1, 1, 1, 4, 1, 1, 1, 1, 1, 4, 1, 4, 1, 1, 1, 1, 1, 1, 1, 6, 1, 1, 1, 1, 1, 1, 1, 4, 1, 1, 1, 1, 1, 1, 1, 4, 4, 1, 1, 1, 1, 1, 1

Offset

1,8

Comments

First differs from A365487 at n = 32.

Links

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

Formula

a(n) = A000005(A368170(n)).

Multiplicative with a(p^e) = 1 if e <= 2, a(p^e) = e+1 if e is odd and e > 1, and a(p^e) = e otherwise.

a(n) >= 1, with equality if and only if n is cubefree (A004709).

a(n) <= A000005(n), with equality if and only if n is in A335988.

Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = zeta(2) * Product_{p prime} (1 - 1/p^2 + 3/p^3 - 1/p^5) = 1.69824776889117043774... .

Mathematica

f[p_, e_] := If[e <= 2, 1, If[EvenQ[e], e, e + 1]]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100]

Prog

(PARI) a(n) = {my(f=factor(n)); prod(i=1, #f~, if(f[i, 2] <= 2, 1, if(!(f[i, 2]%2), f[i, 2], f[i, 2]+1)))};

Crossrefs

Cf. A000005, A013661, A004709, A286324, A365487, A368170.

Sequence in context: A336649 A370703 A365487 * A359594 A368331 A366145

Adjacent sequences: A368169 A368170 A368171 * A368173 A368174 A368175

Keyword

nonn,easy,mult

Author

Amiram Eldar, Dec 14 2023

Status

approved