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A354385
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a(n) is the smallest odd number that has n middle divisors.
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1
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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
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OFFSET
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1,2
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COMMENTS
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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.
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LINKS
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EXAMPLE
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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.
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MATHEMATICA
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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 *)
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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