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A382493 a(n) is the 2-adic valuation of the least number that has exactly n 3-smooth divisors.

%I A382493 #9 Mar 29 2025 04:22:53
%S A382493 0,1,2,1,4,2,6,3,2,4,10,3,12,6,4,3,16,5,18,4,6,10,22,5,4,12,8,6,28,5,
%T A382493 30,7,10,16,6,8,36,18,12,7,40,6,42,10,8,22,46,7,6,9,16,12,52,8,10,7,
%U A382493 18,28,58,9,60,30,8,7,12,10,66,16,22,9,70,11,72,36,14
%N A382493 a(n) is the 2-adic valuation of the least number that has exactly n 3-smooth divisors.
%C A382493 a(n) = d-1, where d is the divisor of n that minimizes 2^(x-1)*3^(n/x-1).
%C A382493 The 3-adic valuation of A382492(n) is A007949(A382492(n)) = n/(a(n)+1) - 1.
%H A382493 Amiram Eldar, <a href="/A382493/b382493.txt">Table of n, a(n) for n = 1..10000</a>
%F A382493 a(n) = A007814(A382492(n)).
%F A382493 a(n) >= A033677(n) - 1 (inequality occurs at n = 36, 72, 120, 144, 180, 210, 240, ...).
%F A382493 a(p) = p-1 for prime p.
%F A382493 a(n) = A099311(n) if n is in A037143.
%t A382493 a[n_] := IntegerExponent[Min[Table[2^(d-1)*3^(n/d-1), {d, Divisors[n]}]], 2]; Array[a, 100]
%o A382493 (PARI) a(n) = valuation(vecmin(apply(d -> 2^(d-1)*3^(n/d-1), divisors(n))), 2);
%Y A382493 Cf. A007814, A007949, A037143, A099311, A382492.
%K A382493 nonn,easy
%O A382493 1,3
%A A382493 _Amiram Eldar_, Mar 29 2025