A368336 The number of divisors of the largest term of A072873 that divides of n.

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

Offset: 1

Author

Amiram Eldar, Dec 21 2023

Keywords

• nonn
• easy
• mult

Links

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

Crossrefs

Cf. A000005, A048103, A072873, A327939, A365632, A368331, A368335.

Programs

Mathematica

f[p_, e_] := (e - Mod[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~, f[i,2] - f[i,2]%f[i,1] + 1);}

Formula

a(n) = A000005(A327939(n)).

Multiplicative with a(p^e) = e - (e mod p) + 1.

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 A072873.

Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = Product_{p prime} (1 + p/(p^p-1)) = 1.86196549645040699446... .