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A090585 Numerator of (Sum_{k=1..n} k) / (Product_{k=1..n} k).

%I A090585 #18 Jul 02 2016 22:56:58
%S A090585 1,3,1,5,1,7,1,1,1,11,1,13,1,1,1,17,1,19,1,1,1,23,1,1,1,1,1,29,1,31,1,
%T A090585 1,1,1,1,37,1,1,1,41,1,43,1,1,1,47,1,1,1,1,1,53,1,1,1,1,1,59,1,61,1,1,
%U A090585 1,1,1,67,1,1,1,71,1,73,1,1,1,1,1,79,1,1,1,83,1
%N A090585 Numerator of (Sum_{k=1..n} k) / (Product_{k=1..n} k).
%C A090585 If the offset is set to 2 then [a(n) <> 1] is the indicator function of the odd primes ([] Iverson bracket). [_Peter Luschny_, Jul 05 2009]
%F A090585 a(n) = A000217(n) / A069268(n).
%F A090585 a(n) = A089026(n+1) for n>1.
%F A090585 Also for n>1, a(n) is a numerator of determinant of (n-1) X (n-1) matrix with M(i,j) = (i+2)/(i+1) if i=j, otherwise 1. E.g., a(2) = Numerator[Det[{{3/2}}]] = Numerator[3/2] = 3. a(3) = Numerator[Det[{{3/2,1},{1,4/3}}]] = Numerator[1/1] = 1. a(4) = Numerator[Det[{{3/2,1,1},{1,4/3,1},{1,1,5/4}}]] = Numerator[5/12] = 5. - _Alexander Adamchuk_, May 26 2006
%F A090585 a(n) = gcd(n*(n+1)/2, n!+1). [_Jaume Oliver Lafont_, Jan 23 2009]
%e A090585 For n=5, (1+2+3+4+5)/(1*2*3*4*5) = 15/120 = 1/8, so a(5) = 1. For n=6, (1+2+3+4+5+6)/(1*2*3*4*5*6) = 21/720 = 7/240, so a(6) = 7. - _Michael B. Porter_, Jul 02 2016
%p A090585 a := n -> denom(2*n!/(n+1)); # _Peter Luschny_, Jul 05 2009
%t A090585 With[{nn=100},Numerator[Accumulate[Range[nn]]/Rest[FoldList[Times,1,Range[nn]]]]] (* _Harvey P. Dale_, Sep 09 2014 *)
%o A090585 (PARI) for(n=1,100,print1(gcd(n*(n+1)/2,round(factorial(n))+1),", ")); \\ _Jaume Oliver Lafont_, Jan 23 2009
%Y A090585 Denominator = A090586.
%Y A090585 Cf. A000217, A069268, A089026, A118679.
%K A090585 nonn,frac
%O A090585 1,2
%A A090585 _Reinhard Zumkeller_, Dec 03 2003