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

1,3

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

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.

Mathematica

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

Prog

(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)); }

Crossrefs

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

Sequence in context: A238580 A035176 A339258 * A322976 A011793 A109649

Adjacent sequences: A360323 A360324 A360325 * A360327 A360328 A360329

Keyword

nonn,mult

Author

Amiram Eldar, Feb 03 2023

Status

approved