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%I #13 Oct 24 2015 00:11:46
%S 2,6,4,30,3,84,8,90,20,11,12,5460,7,40,48,1530,9,7980,20,1155,88,276,
%T 24,81900,78,189,35,1160,15,38192,32,16830,51,315,72,3838380,19,780,
%U 280,142065,21,132440,44,828,5520,376,48,9746100,200,14586
%N Denominator of the rational number Sum_{d|n}1/(d+1).
%C Conjecture: For any positive integers k and s, all the numbers Sum_{d|n}1/(d+k)^s (n = 1,2,3,...) have pairwise distinct fractional parts, and none of them is an integer.
%C This implies that a(n) > 1 for all n > 0.
%C See also A001157 for a similar conjecture involving Sum_{d|n}1/d^s.
%C I have verified that Sum_{d|n}1/(d+1) (n = 1..2*10^5) indeed have pairwise distinct fractional parts and none of them is an integer. For each k = 2,3,4,5,6 I have verified that Sum_{d|n}1/(d+k) (n = 1..10^5) have pairwise distinct fractional parts and none of them is integral. - _Zhi-Wei Sun_, Oct 20 2015.
%H Zhi-Wei Sun, <a href="/A263326/b263326.txt">Table of n, a(n) for n = 1..10000</a>
%e a(1) = 2 since sum_{d|1}1/(d+1) = 1/2.
%e a(2) = 6 since sum_{d|2}1/(d+1) = 1/2 + 1/3 = 5/6.
%e a(3) = 4 since sum_{d|3}1/(d+1) = 1/2 + 1/4 = 3/4.
%p f:= n -> denom(add(1/(d+1),d=numtheory:-divisors(n))):
%p map(f, [$1..100]); # _Robert Israel_, Oct 20 2015
%t Dv[n_]:=Dv[n]=Divisors[n]
%t a[n_]:=a[n]=Denominator[Sum[1/(Part[Dv[n],i]+1),{i,1,Length[Dv[n]]}]]
%t Do[Print[n," ",a[n]],{n,1,50}]
%o (PARI) a(n) = denominator(sumdiv(n, d, 1/(d+1))); \\ _Michel Marcus_, Oct 15 2015
%Y Cf. A000203, A001157, A263317, A263319, A263325.
%K nonn
%O 1,1
%A _Zhi-Wei Sun_, Oct 14 2015
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