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

1, 3, 6, 15, 18, 36, 30, 105, 60, 120, 90, 315, 816, 1360, 180, 700, 450, 360, 720, 1008, 420, 1540, 630, 900, 840, 1080, 1620, 1680, 2160, 1800, 1890, 5280, 1260, 3240, 3150, 17325, 7200, 29120, 5670, 9072, 2520, 3960, 10296, 18144, 3780, 20020, 5040, 7920, 10800

Offset

1,2

Comments

From Peter Munn, May 15 2025 and May 20 2025: (Start)

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.

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.)

(End)

If a(n) = m then m has at least n divisors. - David A. Corneth, May 16 2025

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

Links

David A. Corneth, Table of n, a(n) for n = 1..872

David A. Corneth, PARI program

David A. Corneth, Upper bounds on a(n) for n = 1..10000

Michael De Vlieger, Prime Power Decomposition of A383402(n), n = 1..261.

Formula

a(n) = min({k : A000005(k) >= n & A027750(k,n) = A000265(k)}). - Peter Munn, May 14 2025

Example

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.

Mathematica

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 *)

Prog

(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
(PARI) \\ See Corneth link

Crossrefs

Row 1 of A383961.

The range of terms is a subset of {1} U A355200.

Cf. A000005, A000265, A001227, A005117, A027750, A038547, A064989, A182469, A383401.

See A221647 for other sequences giving the smallest number whose n-th divisor satisfies some condition.

Sequence in context: A160724 A212060 A249246 * A248969 A174279 A233554

Adjacent sequences: A383399 A383400 A383401 * A383403 A383404 A383405

Keyword

nonn

Author

Omar E. Pol, May 14 2025

Extensions

More terms from Amiram Eldar, May 14 2025

Status

approved