A368331 The number of divisors of the largest term of A054743 that divides of n.

1, 1, 1, 1, 1, 1, 1, 4, 1, 1, 1, 1, 1, 1, 1, 5, 1, 1, 1, 1, 1, 1, 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, 5, 1, 1, 1, 1, 1, 1, 1, 4, 1, 1, 1, 1, 1, 1, 1, 7, 1, 1, 1, 1, 1, 1, 1, 4, 1, 1, 1, 1, 1, 1, 1, 5, 5, 1, 1, 1, 1, 1, 1

Offset

1,8

Comments

First differs from A366145 at n = 27.

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(A368329(n)).

a(n) >= 1, with equality if and only if n is in A207481.

a(n) <= A000005(n), with equality if and only if n is in A054743.

Dirichlet g.f.: zeta(s)^2 * Product_{p prime} (1 - 1/p^s - 1/p^((p+2)*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))) = 1.58396891058853238595... .

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, A054743, A207481, A368328, A368329, A368330, A368335, A368336.

Cf. A366145.

Sequence in context: A365487 A368172 A359594 * A366145 A204160 A360165

Adjacent sequences: A368328 A368329 A368330 * A368332 A368333 A368334

Keyword

nonn,easy,mult

Author

Amiram Eldar, Dec 21 2023

Status

approved