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A265393 a(n) = the smallest number k such that floor(Sum_{d|k} 1/tau(d)) = n.

%I A265393 #46 Sep 08 2022 08:46:14
%S A265393 1,6,24,60,180,420,840,2520,4620,9240,13860,27720,60060,55440,110880,
%T A265393 166320,180180,480480,360360,900900,720720,1441440,1801800,2162160,
%U A265393 3063060,4084080,7207200,12612600,6126120,27027000,12252240,18378360,43243200,24504480
%N A265393 a(n) = the smallest number k such that floor(Sum_{d|k} 1/tau(d)) = n.
%C A265393 Further known terms: a(29) = 6126120, a(31) = 12252240.
%C A265393 Are there numbers n > 1 such that Sum_{d|n} 1/tau(d) is an integer?
%C A265393 Sequences of numbers n such that floor(Sum_{d|n} 1/tau(d)) = k for k = 1..6:
%C A265393 k=1: 1, 2, 3, 4, 5, 7, 9, 11, 13, 17, 19, 23, 25, 29, 31, 37, 41, ... (A166684);
%C A265393 k=2: 6, 8, 10, 12, 14, 15, 16, 18, 20, 21, 22, 26, 27, 28, 32, 33, 34, 35, ...;
%C A265393 k=3: 24, 30, 36, 40, 42, 48, 54, 56, 66, 70, 72, 78, 80, 88, 96, 100, ...;
%C A265393 k=4: 60, 84, 90, 120, 126, 132, 140, 144, 150, 156, 168, 198, 204, 216, ...;
%C A265393 k=5: 180, 210, 240, 252, 300, 330, 336, 360, 390, 396, 450, 462, 468, ...;
%C A265393 k=6: 420, 630, 660, 720, 780, 900, 924, 990, 1008, 1020, 1050, 1080, ....
%C A265393 Values of function F = Sum_{d|n} 1/tau(d) for some numbers according to their prime signature: F{} = 1; F{1} = 3/2; F{2} = 11/6; F{1, 1} = 9/4; F{3} = 25/12; F{2, 1} = 11/4; F{4} = 137/60; F{3, 1} = 25/8, ...
%H A265393 Hiroaki Yamanouchi, <a href="/A265393/b265393.txt">Table of n, a(n) for n = 1..5000</a>
%e A265393 For n = 2; a(2) = 6 because 6 is the smallest number with floor(Sum_{d|6} 1/tau(d)) = floor(1/1 + 1/2 + 1/2 + 1/4) = floor(9/4) = 2.
%t A265393 Table[k = 1; While[Floor@ Sum[1/DivisorSigma[0, d], {d, Divisors@ k}] != n, k++]; k, {n, 17}] (* _Michael De Vlieger_, Dec 09 2015 *)
%o A265393 (Magma) a:=1; S:=[a]; for n in [2..14] do k:=0; flag:= true; while flag do k+:=1; if Floor(&+[1/NumberOfDivisors(d): d in Divisors(k)]) eq n then Append(~S, k); a:=k; flag:=false; end if; end while; end for; S;
%o A265393 (PARI) a(n) = {k=1; while(k, if(floor(sumdiv(k, d, 1/numdiv(d))) == n, return(k)); k++)} \\ _Altug Alkan_, Dec 09 2015
%Y A265393 Cf. A000005, A253139, A265390, A265391, A265392.
%Y A265393 Cf. A237350 (a(n) = the smallest number k such that Sum_{d|k} 1/tau(d) >= n).
%K A265393 nonn
%O A265393 1,2
%A A265393 _Jaroslav Krizek_, Dec 08 2015
%E A265393 More terms from _Michel Marcus_, Dec 23 2015
%E A265393 a(33)-a(34) from _Hiroaki Yamanouchi_, Dec 31 2015