A354385 a(n) is the smallest odd number that has n middle divisors.

1, 15, 1225, 2145, 99225, 17955, 893025, 51975, 4601025, 315315, 16769025, 855855, 12006225, 2567565, 108056025, 6531525, 385533225, 11486475, 225450225, 16787925, 1329696225, 38513475, 2701400625, 77702625, 6053618025, 80405325, 4846248225, 101846745, 2029052025, 218243025

Offset

1,2

Comments

This sequence is nonincreasing since a(5) > a(6), neither is the subsequence a(2n-1), n >= 1, of record odd counts of middle divisors since a(11) = 16769025 > 12006225 = a(13), nor is the subsequence a(2n), n >= 1, of record even counts since a(32) = 413377965 > 334639305 = a(34).

a(21) > 5*10^8.

Further computed values at even indices up to 5*10^8 are a(22, 24, 26, 28, 30, 32, 34) = (38513475, 77702625, 80405325, 101846745, 218243025, 413377965, 334639305).

Observation: At present all known terms >= a(4) are divisible by 3, all >= a(10) are divisible by 7, all >= a(12) are divisible by 11.

Conjecture: For every k, there is an n such that all >= a(n) are divisible by the first k odd primes.

Links

Table of n, a(n) for n=1..30.

Omar E. Pol, Comments on A354385.

Example

a(2) = 15 = A319529(3) is the smallest odd number with 2 middle divisors: 3 and 5.
a(3) = 1225 = A319529(116) is the smallest odd number with 3 middle divisors: 25, 35, and 45.

Mathematica

middleDivC[n_] := Length[Select[Divisors[n], Sqrt[n/2]<=#<Sqrt[2n]&]]
(* parameter b estimates the number of middle divisor counts for range 1...n *)
a354385[n_, b_] := Module[{list=Table[0, b], k, c}, For[k=1, k<=n, k+=2, c=middleDivC[k]; If[c>=1&&list[[c]]==0, list[[c]]=k]]; list]
a354385[2*10^7, 20] (* long computation time *)

Crossrefs

Cf. A067742, A016754, A128605, A235791, A237048, A237593, A245092, A249223, A319529, A320137.

Sequence in context: A211901 A059383 A206394 * A098723 A120296 A209679

Adjacent sequences: A354382 A354383 A354384 * A354386 A354387 A354388

Keyword

nonn

Author

Hartmut F. W. Hoft, May 24 2022

Extensions

More terms from Amiram Eldar, Jun 07 2022

Edited by Omar E. Pol at the suggestion of N. J. A. Sloane, Jul 28 2022

Status

approved