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

1,8

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

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... .

Mathematica

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

Crossrefs

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

Sequence in context: A204127 A225174 A059895 * A321167 A190867 A355936

Adjacent sequences: A368325 A368326 A368327 * A368329 A368330 A368331

Keyword

nonn,easy,mult

Author

Amiram Eldar, Dec 21 2023

Status

approved