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

%I A185445 #20 Aug 18 2024 06:31:52
%S A185445 24,60,180,240,900,960,720,2880,15360,3600,6480,61440,14400,46080,
%T A185445 983040,25920,32400,3932160,184320,62914560,233280,230400,2949120,
%U A185445 129600,414720,11796480,4026531840,921600,16106127360,810000,1658880,188743680,1166400,1030792151040,14745600,3732480
%N A185445 Smallest number having exactly t divisors, where t is the n-th triprime (A014612).
%C A185445 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).
%C A185445 The first row is A061286, the second row is A096932.
%F A185445 a(n) = A005179(A014612(n)).
%e A185445 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.
%e A185445 a(20) is 62914560 because the 10th triprime is 92, and the smallest number with exactly 92 factors is 62914560 = 2^22 * 3 * 5.
%o A185445 (Python)
%o A185445 from math import isqrt, prod
%o A185445 from sympy import isprime, primepi, primerange, integer_nthroot, prime, divisors
%o A185445 def A185445(n):
%o A185445     def mult_factors(n):
%o A185445         if isprime(n):
%o A185445             return [(n,)]
%o A185445         c = []
%o A185445         for d in divisors(n,generator=True):
%o A185445             if 1<d<n:
%o A185445                 for a in mult_factors(n//d):
%o A185445                     c.append(tuple(sorted((d,)+a)))
%o A185445         return list(set(c))
%o A185445     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)))
%o A185445     m, k = n, f(n)
%o A185445     while m != k:
%o A185445         m, k = k, f(k)
%o A185445     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
%Y A185445 Cf. A005179, A014612, A061286, A096932.
%K A185445 nonn,easy
%O A185445 1,1
%A A185445 _Jonathan Vos Post_, Feb 03 2011