A368332 The number of terms of A054744 that divide n.

1, 1, 1, 2, 1, 1, 1, 3, 1, 1, 1, 2, 1, 1, 1, 4, 1, 1, 1, 2, 1, 1, 1, 3, 1, 1, 2, 2, 1, 1, 1, 5, 1, 1, 1, 2, 1, 1, 1, 3, 1, 1, 1, 2, 1, 1, 1, 4, 1, 1, 1, 2, 1, 2, 1, 3, 1, 1, 1, 2, 1, 1, 1, 6, 1, 1, 1, 2, 1, 1, 1, 3, 1, 1, 1, 2, 1, 1, 1, 4, 3, 1, 1, 2, 1, 1, 1

Offset

1,4

Comments

The number of divisors d of n such that e >= p for all prime powers p^e in the prime factorization of d (i.e., e >= 1 and p^(e+1) does not divide d).

The largest of these divisors is A368333(n).

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 - p + 2 if e >= p.

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

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

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 - p + 2]; 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] - f[i, 1] + 2)); }

Crossrefs

Cf. A054744, A365632, A368328, A368333, A368334, A368335.

Sequence in context: A079229 A344972 A204988 * A224765 A369167 A160267

Adjacent sequences: A368329 A368330 A368331 * A368333 A368334 A368335

Keyword

nonn,easy,mult

Author

Amiram Eldar, Dec 21 2023

Status

approved