A377516 The number of divisors of n that are terms in A276078.

1, 2, 2, 2, 2, 4, 2, 2, 3, 4, 2, 4, 2, 4, 4, 2, 2, 6, 2, 4, 4, 4, 2, 4, 3, 4, 3, 4, 2, 8, 2, 2, 4, 4, 4, 6, 2, 4, 4, 4, 2, 8, 2, 4, 6, 4, 2, 4, 3, 6, 4, 4, 2, 6, 4, 4, 4, 4, 2, 8, 2, 4, 6, 2, 4, 8, 2, 4, 4, 8, 2, 6, 2, 4, 6, 4, 4, 8, 2, 4, 3, 4, 2, 8, 4, 4, 4

Offset: 1

Author

Amiram Eldar, Oct 30 2024

Keywords

• nonn
• easy
• mult

Comments

The sum of these divisors is A377517(n), and the largest of them is A377515(n).

Links

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

Crossrefs

Cf. A000005, A000720, A013661, A276078, A377515, A377517, A377519.

Programs

Mathematica

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

PARI

a(n) = {my(f = factor(n)); prod(i = 1, #f~, min(primepi(f[i,1]), f[i,2]) + 1);}

Formula

a(n) = A000005(A377515(n)).

Multiplicative with a(p^e) = min(pi(p), e) + 1, where pi(n) = A000720(n).

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