A360326 a(n) is the number of divisors of n that have only prime-indexed prime factors.

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

Offset: 1

Author

Amiram Eldar, Feb 03 2023

Keywords

• nonn
• mult

Comments

First differs from A322976 at n = 21.

Equivalently, a(n) is the number of divisors of the largest divisor of n that has only prime-indexed prime factors.

The asymptotic mean of this sequence is Product_{p in A006450} p/(p-1) > 3. See A076610 for a numerical estimate of the value of this product.

Links

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

Crossrefs

Cf. A000005, A006450, A076610, A320628, A322976, A360325, A360327.

Programs

Mathematica

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

PARI

a(n) = {my(f = factor(n), p = f[,1], e = f[,2]); prod(i = 1, #p, if(isprime(primepi(p[i])), e[i]+1, 1));}

Formula

a(n) = 1 if and only if n is in A320628.

a(n) = A000005(n) if and only if n is in A076610.

a(n) = A000005(A360325(n)).

Multiplicative with a(p^e) = e+1 if p is a prime-indexed prime (A006450), and 1 otherwise.