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

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, 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, 1, 1, 1, 67, 1, 1, 1, 71, 1, 73, 1, 1, 1, 1, 1, 79, 1, 1, 1, 83, 1

Offset: 1

Author

Reinhard Zumkeller, Dec 03 2003

Keywords

• nonn
• frac

Comments

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]

Examples

			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
		

Crossrefs

Denominator = A090586.

Cf. A000217, A069268, A089026, A118679.

Programs

Maple

a := n -> denom(2*n!/(n+1)); # Peter Luschny, Jul 05 2009

Mathematica

With[{nn=100},Numerator[Accumulate[Range[nn]]/Rest[FoldList[Times,1,Range[nn]]]]] (* Harvey P. Dale, Sep 09 2014 *)

PARI

for(n=1,100,print1(gcd(n*(n+1)/2,round(factorial(n))+1),", ")); \\ Jaume Oliver Lafont, Jan 23 2009

Formula

a(n) = A000217(n) / A069268(n).

a(n) = A089026(n+1) for n>1.

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

a(n) = gcd(n*(n+1)/2, n!+1). [Jaume Oliver Lafont, Jan 23 2009]