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

Author

Hartmut F. W. Hoft, May 24 2022

Keywords

• nonn

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.

Examples

			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.
		

Links

• Omar E. Pol, Comments on A354385.

Crossrefs

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

Programs

Mathematica

middleDivC[n_] := Length[Select[Divisors[n], Sqrt[n/2]<=#=1&&list[[c]]==0, list[[c]]=k]]; list]
a354385[2*10^7, 20] (* long computation time *)

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