A368328 The number of terms of A054743 that divide n.

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

Offset: 1

Author

Amiram Eldar, Dec 21 2023

Keywords

• nonn
• easy
• mult

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 A368329(n).

Links

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

Crossrefs

Cf. A054743, A207481, A365632, A368329, A368330, A368331, A368332.

Programs

Mathematica

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

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] + 1));}

Formula

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

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

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

Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = Product_{p prime} (1 + 1/((p-1)*p^p)) = 1.27325025767774256043... .