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Smallest number having exactly s divisors, where s is the n-th semiprime (A001358).
13

%I #18 Apr 13 2024 05:13:21

%S 6,12,36,48,192,144,576,3072,1296,12288,9216,196608,5184,786432,36864,

%T 12582912,46656,589824,82944,2359296,805306368,3221225472,331776,

%U 37748736,206158430208,746496,3298534883328,5308416,13194139533312,2415919104,2985984,9663676416

%N Smallest number having exactly s divisors, where s is the n-th semiprime (A001358).

%C This is to smallest integer for which the number of divisors is the n-th prime (A061286) as semiprimes (A001358) are to primes (A000040). - _Jonathan Vos Post_, Feb 03 2011

%H Amiram Eldar, <a href="/A096932/b096932.txt">Table of n, a(n) for n = 1..1000</a>

%F A000005(a(n)) = A001358(n) and A000005(m) <> A001358(n) for m < a(n).

%F a(n) = A005179(A001358(n)).

%F a(p*q) = 2^(q-1) * 3^(p-1) for primes p <= q.

%F a(A000040(i)*A000040(j)) = 2^(A084127(j)-1) * 3^(A084127(i)-1) for i <= j.

%t s[n_] := Module[{f = FactorInteger[n], p, q}, If[Total[f[[;;,2]]] == 2, p=f[[1,1]]; q = n/p; 2^(q-1) * 3^(p-1) ,Nothing]]; Array[s, 100] (* _Amiram Eldar_, Apr 13 2024 *)

%Y Cf. A000005, A001358, A005179, A061234, A061286, A061299, A084127.

%K nonn

%O 1,1

%A _Reinhard Zumkeller_, Jul 15 2004