OFFSET
1,1
COMMENTS
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
KEYWORD
nonn,easy
AUTHOR
Jonathan Vos Post, Feb 03 2011
STATUS
approved