A185445 Smallest number having exactly t divisors, where t is the n-th triprime (A014612).

24, 60, 180, 240, 900, 960, 720, 2880, 15360, 3600, 6480, 61440, 14400, 46080, 983040, 25920, 32400, 3932160, 184320, 62914560, 233280, 230400, 2949120, 129600, 414720, 11796480, 4026531840, 921600, 16106127360, 810000, 1658880, 188743680, 1166400, 1030792151040, 14745600, 3732480

Offset

1,1

Comments

This is the 3rd row of an infinite array A[k,n] = smallest number having exactly j divisors where j is the n-th natural number with exactly k prime factors (with multiplicity).

The first row is A061286, the second row is A096932.

Links

Table of n, a(n) for n=1..36.

Formula

a(n) = A005179(A014612(n)).

Example

a(10) is 3600 because the 10th triprime is 45, and the smallest number with exactly 45 factors is 3600 = 2^4 * 3^2 * 5^2.
a(20) is 62914560 because the 10th triprime is 92, and the smallest number with exactly 92 factors is 62914560 = 2^22 * 3 * 5.

Prog

(Python)
from math import isqrt, prod
from sympy import isprime, primepi, primerange, integer_nthroot, prime, divisors
def A185445(n):
    def mult_factors(n):
        if isprime(n):
            return [(n, )]
        c = []
        for d in divisors(n, generator=True):
            if 1<d<n:
                for a in mult_factors(n//d):
                    c.append(tuple(sorted((d, )+a)))
        return list(set(c))
    def f(x): return int(n+x-sum(primepi(x//(k*m))-b for a, k in enumerate(primerange(integer_nthroot(x, 3)[0]+1)) for b, m in enumerate(primerange(k, isqrt(x//k)+1), a)))
    m, k = n, f(n)
    while m != k:
        m, k = k, f(k)
    return min((prod(prime(i)**(j-1) for i, j in enumerate(reversed(d), 1)) for d in mult_factors(m)), default=1) # Chai Wah Wu, Aug 17 2024

Crossrefs

Cf. A005179, A014612, A061286, A096932.

Sequence in context: A240524 A250577 A242844 * A321840 A052759 A356911

Adjacent sequences: A185442 A185443 A185444 * A185446 A185447 A185448

Keyword

nonn,easy

Author

Jonathan Vos Post, Feb 03 2011

Status

approved