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

1,1

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

Formula

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

Mathematica

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

Prog

(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

Crossrefs

Cf. A000005, A009087, A141241.

Sequence in context: A366922 A218800 A062502 * A130451 A130285 A046027

Adjacent sequences: A141239 A141240 A141241 * A141243 A141244 A141245

Keyword

nonn,changed

Author

Leroy Quet, Jun 16 2008

Extensions

Extended by Ray Chandler, Jun 25 2009

Status

approved