A368335 The number of divisors of the largest term of A054744 that divides of n.

1, 1, 1, 3, 1, 1, 1, 4, 1, 1, 1, 3, 1, 1, 1, 5, 1, 1, 1, 3, 1, 1, 1, 4, 1, 1, 4, 3, 1, 1, 1, 6, 1, 1, 1, 3, 1, 1, 1, 4, 1, 1, 1, 3, 1, 1, 1, 5, 1, 1, 1, 3, 1, 4, 1, 4, 1, 1, 1, 3, 1, 1, 1, 7, 1, 1, 1, 3, 1, 1, 1, 4, 1, 1, 1, 3, 1, 1, 1, 5, 5, 1, 1, 3, 1, 1, 1

Offset

1,4

Links

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

Formula

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

a(n) = A000005(A368333(n)).

a(n) >= 1, with equality if and only if n is in A048103.

a(n) <= A000005(n), with equality if and only if n is in A054744.

Dirichlet g.f.: zeta(s)^2 * Product_{p prime} (1 - 1/p^s + 1/p^(p*s-1) + 1/p^((p+1)*s) - 1/p^((p+1)*s-1)).

Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = Product_{p prime} (1 + (1 + (p-1)*p)/((p-1)*p^p)) = 1.98019019497523582894... .

Mathematica

f[p_, e_] := If[e < p, 1, 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] < f[i, 1], 1, f[i, 2]+1)); }

Crossrefs

Cf. A000005, A048103, A054744, A368331, A368332, A368333, A368334, A368336.

Sequence in context: A336455 A130605 A354872 * A157261 A079110 A079619

Adjacent sequences: A368332 A368333 A368334 * A368336 A368337 A368338

Keyword

nonn,easy,mult

Author

Amiram Eldar, Dec 21 2023

Status

approved