A141242 a(n) is the number of divisors of the n-th positive integer with a prime number of divisors. In other words, a(n) is the number of divisors of A009087(n).

2, 2, 3, 2, 2, 3, 2, 2, 5, 2, 2, 2, 3, 2, 2, 2, 2, 2, 2, 3, 2, 2, 2, 7, 2, 2, 2, 2, 5, 2, 2, 2, 2, 2, 2, 2, 2, 3, 2, 2, 2, 2, 2, 2, 2, 2, 2, 3, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 3, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 3, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2

Offset: 1

Author

Leroy Quet, Jun 16 2008

Keywords

• nonn

Comments

A009087(n) is of the form p^(a(n)-1), where p is some prime.

Links

• Michael De Vlieger, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A000005, A009087, A141241.

Programs

Mathematica

DivisorSigma[0, #] &@ Select[Range@ 500, PrimeQ@ DivisorSigma[0, #] &] (* Michael De Vlieger, Aug 19 2017 *)

Python

from sympy import primepi, integer_nthroot, primerange, factorint
def A141242(n):
    def bisection(f,kmin=0,kmax=1):
        while f(kmax) > kmax: kmax <<= 1
        kmin = kmax >> 1
        while kmax-kmin > 1:
            kmid = kmax+kmin>>1
            if f(kmid) <= kmid:
                kmax = kmid
            else:
                kmin = kmid
        return kmax
    def f(x): return int(n+x-sum(primepi(integer_nthroot(x,k-1)[0]) for k in primerange(x.bit_length()+1)))
    return list(factorint(bisection(f,n,n)).values())[0]+1 # Chai Wah Wu, Feb 22 2025

Formula

a(n) = A000005(A009087(n)).

Extensions

Extended by Ray Chandler, Jun 25 2009