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A383402 Smallest number whose largest odd divisor is its n-th divisor.

%I A383402 #55 Jun 01 2025 09:57:15
%S A383402 1,3,6,15,18,36,30,105,60,120,90,315,816,1360,180,700,450,360,720,
%T A383402 1008,420,1540,630,900,840,1080,1620,1680,2160,1800,1890,5280,1260,
%U A383402 3240,3150,17325,7200,29120,5670,9072,2520,3960,10296,18144,3780,20020,5040,7920,10800
%N A383402 Smallest number whose largest odd divisor is its n-th divisor.
%C A383402 From _Peter Munn_, May 15 2025 and May 20 2025: (Start)
%C A383402 A038547 is easily seen to be an upper bound for the sequence and a term equals this upper bound if and only if it is odd. Moreover, if a(n) = 2m with m odd, then the largest odd divisor of 2m is m, its second largest divisor, and a(n) = 2 * A038547((n+1)/2). It follows that 1 is the only term not divisible by 4 or by a nonunit term of A038547.
%C A383402 a(8) = 105 is the last squarefree term. (This is a corollary to lemma: prime p > 9 cannot be a divisor of a squarefree term. Proof of lemma: Let p divide squarefree k. If 3p is also divisor, set m = 9k/p, otherwise set m = 3k/p. Then k is not a term as m is a smaller number whose largest odd divisor is in the same position in the divisor list.)
%C A383402 (End)
%C A383402 If a(n) = m then m has at least n divisors. - _David A. Corneth_, May 16 2025
%C A383402 Every term a(n) = t > 1 is divisible by 2 or 3. Proof: Suppose it is not. Then it is odd and n is the number of divisors of t (cf. A000005). But t is not the smallest number that has n odd divisors that is odd. Setting every prime factor p to the largest prime < p and then multiplying gives a smaller odd number that has n divisors (cf. A064989). - _David A. Corneth_, May 17 2025
%H A383402 David A. Corneth, <a href="/A383402/b383402.txt">Table of n, a(n) for n = 1..872</a>
%H A383402 David A. Corneth, <a href="/A383402/a383402_1.gp.txt">PARI program</a>
%H A383402 David A. Corneth, <a href="/A383402/a383402_2.gp.txt">Upper bounds on a(n) for n = 1..10000</a>
%H A383402 Michael De Vlieger, <a href="/A383402/a383402.txt">Prime Power Decomposition of A383402(n), n = 1..261</a>.
%F A383402 a(n) = min({k : A000005(k) >= n & A027750(k,n) = A000265(k)}). - _Peter Munn_, May 14 2025
%e A383402 The divisors of 18 are [1, 2, 3, 6, 9, 18] and the largest odd divisor is 9 and 9 is its 5th divisor, so a(5) = 18 because 18 the smallest number having that property.
%t A383402 With[{t = Table[If[OddQ[n], DivisorSigma[0, n], FirstPosition[Divisors[n], n/2^IntegerExponent[n, 2]][[1]]], {n, 1, 30000}]}, TakeWhile[FirstPosition[t, #] & /@ Range[Max[t]] // Flatten, ! MissingQ[#] &]] (* _Amiram Eldar_, May 14 2025 *)
%o A383402 (PARI) a(n) = my(k=1); while (select(x->(x==k/2^valuation(k,2)), divisors(k), 1)[1] != n, k++); k; \\ _Michel Marcus_, May 14 2025
%o A383402 (PARI) \\ See Corneth link
%Y A383402 Row 1 of A383961.
%Y A383402 The range of terms is a subset of {1} U A355200.
%Y A383402 Cf. A000005, A000265, A001227, A005117, A027750, A038547, A064989, A182469, A383401.
%Y A383402 See A221647 for other sequences giving the smallest number whose n-th divisor satisfies some condition.
%K A383402 nonn
%O A383402 1,2
%A A383402 _Omar E. Pol_, May 14 2025
%E A383402 More terms from _Amiram Eldar_, May 14 2025