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A265714 a(n) = the smallest number k such that floor(Sum_{d|k} 1/sigma(d)) = n.

%I A265714 #23 Sep 08 2022 08:46:15
%S A265714 1,60,110880,4658179125600,950542574818669103079134726400,
%T A265714 204614292026733833316841991529248485168966921782532186656980932752000
%N A265714 a(n) = the smallest number k such that floor(Sum_{d|k} 1/sigma(d)) = n.
%C A265714 If a(4) exists, it must be bigger than 5*10^6.
%C A265714 Are there numbers n > 1 such that Sum_{d|n} 1/sigma(d) is an integer?
%C A265714 Conjecture: a(n) is also the smallest number k such that Sum_{d|k} 1/sigma(d) >= n.
%C A265714 Sequence of numbers n such that floor(Sum_{d|n} 1/sigma(d)) = k for k = 1, 2, 3:
%C A265714 k = 1: 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, ... (A265711);
%C A265714 k = 2: 60, 72, 84, 90, 120, 144, 168, 180, 210, 216, 240, 252, ... (A265712);
%C A265714 k = 3: 110880, 166320, 221760, 277200, 327600, 332640, 360360, ... (A265713).
%C A265714 From _Robert Israel_, Dec 24 2015: (Start)
%C A265714 Note that g(k) = Sum_{d | k} 1/sigma(d) is multiplicative, with g(p) = 1 + 1/(p+1) for prime p.
%C A265714 Since Product_p (1+1/(p+1)) diverges, there are certainly numbers k with floor(g(k)) = n, including some squarefree numbers.
%C A265714 Conjectured values: a(4) = 4658179125600,
%C A265714   a(5) = 1188178218523336378848918408000,
%C A265714   a(6) = 5354073974699535305124032111682002028587967786642925550857667740344000.
%C A265714 These do have the correct value of floor(g(k)), but may not be the lowest possible.
%C A265714 (End)
%C A265714 Probably, a(7) = 1058687979...2471360000 = 349# * 23# * 7# * 5# * 3#^2 * 2#^3. - _Hiroaki Yamanouchi_, Dec 31 2015
%e A265714 For n = 2; a(2) = 60 because 60 is the smallest number with floor (Sum_{d|60} 1/sigma(d)) = floor(155/72) = 2.
%o A265714 (Magma) a:=1; S:=[a]; for n in [2..3] do k:=0; flag:= true; while flag do k+:=1; if &+[1/SumOfDivisors(d): d in Divisors(k)] ge n then Append(~S, k); a:=k; flag:=false; end if; end while; end for; S;
%Y A265714 Cf. A069934, A000203, A265708, A265709, A265710, A265711, A265712, A265713, A266227, A266228.
%K A265714 nonn
%O A265714 1,2
%A A265714 _Jaroslav Krizek_, Dec 24 2015
%E A265714 a(4)-a(6) from _Hiroaki Yamanouchi_, Dec 31 2015