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A061218 Least number whose number of divisors is n-th term from A014613 (numbers of form p*q*r*s, products of exactly 4 primes, counted with multiplicity).

%I A061218 #13 Aug 17 2024 22:29:12
%S A061218 120,360,1260,1680,6300,6720,5040,44100,20160,107520,25200,45360,
%T A061218 430080,100800,322560,176400,6881280,181440,226800,27525120,1290240,
%U A061218 440401920,705600,1632960,1612800,20643840,907200,2903040,1587600,82575360,28185722880,6451200,112742891520
%N A061218 Least number whose number of divisors is n-th term from A014613 (numbers of form p*q*r*s, products of exactly 4 primes, counted with multiplicity).
%F A061218 a(n) = A005179(A014613(n)).
%e A061218 p*q*r*s = 210 is the 27th term in A014613; the smallest number with 210 divisors is 907200 = 2*2*2*2*2*2*3*3*3*3*5*5*7.
%o A061218 (Python)
%o A061218 from math import prod, isqrt
%o A061218 from sympy import primepi, primerange, integer_nthroot, isprime, divisors, prime
%o A061218 def A061218(n):
%o A061218     def f(x): return int(n+x-sum(primepi(x//(k*m*r))-c for a,k in enumerate(primerange(integer_nthroot(x,4)[0]+1)) for b,m in enumerate(primerange(k,integer_nthroot(x//k,3)[0]+1),a) for c,r in enumerate(primerange(m,isqrt(x//(k*m))+1),b)))
%o A061218     def mult_factors(n):
%o A061218         if isprime(n):
%o A061218             return [(n,)]
%o A061218         c = []
%o A061218         for d in divisors(n,generator=True):
%o A061218             if 1<d<n:
%o A061218                 for a in mult_factors(n//d):
%o A061218                     c.append(tuple(sorted((d,)+a)))
%o A061218         return list(set(c))
%o A061218     m, k = n, f(n)
%o A061218     while m != k:
%o A061218         m, k = k, f(k)
%o A061218     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 A061218 Cf. A000005, A005179, A014613, A061148, A061149, A061299.
%K A061218 nonn
%O A061218 1,1
%A A061218 _Labos Elemer_, Jun 06 2001
%E A061218 Corrected and extended by _Michel Marcus_, Sep 05 2017