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

%I A354385 #61 Mar 22 2024 19:00:23
%S A354385 1,15,1225,2145,99225,17955,893025,51975,4601025,315315,16769025,
%T A354385 855855,12006225,2567565,108056025,6531525,385533225,11486475,
%U A354385 225450225,16787925,1329696225,38513475,2701400625,77702625,6053618025,80405325,4846248225,101846745,2029052025,218243025
%N A354385 a(n) is the smallest odd number that has n middle divisors.
%C A354385 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).
%C A354385 a(21) > 5*10^8.
%C A354385 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).
%C A354385 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.
%C A354385 Conjecture: For every k, there is an n such that all >= a(n) are divisible by the first k odd primes.
%H A354385 Omar E. Pol, <a href="/A354385/a354385.txt">Comments on A354385</a>.
%e A354385 a(2) = 15 = A319529(3) is the smallest odd number with 2 middle divisors: 3 and 5.
%e A354385 a(3) = 1225 = A319529(116) is the smallest odd number with 3 middle divisors: 25, 35, and 45.
%t A354385 middleDivC[n_] := Length[Select[Divisors[n], Sqrt[n/2]<=#<Sqrt[2n]&]]
%t A354385 (* parameter b estimates the number of middle divisor counts for range 1...n *)
%t A354385 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]
%t A354385 a354385[2*10^7, 20] (* long computation time *)
%Y A354385 Cf. A067742, A016754, A128605, A235791, A237048, A237593, A245092, A249223, A319529, A320137.
%K A354385 nonn
%O A354385 1,2
%A A354385 _Hartmut F. W. Hoft_, May 24 2022
%E A354385 More terms from _Amiram Eldar_, Jun 07 2022
%E A354385 Edited by _Omar E. Pol_ at the suggestion of _N. J. A. Sloane_, Jul 28 2022