A067857 Sum_{k|n} a(k)/k! = Sum_{j=1 to n} 1/j, sum on left is over positive divisors k of n.

1, 1, 5, 14, 154, 84, 8028, 25584, 361296, 528480, 80627040, 33471360, 13575738240, 13835646720, 263577888000, 13869128448000, 867718162483200, 316745643110400, 309920046408806400, 207862451693568000

Offset

1,3

Comments

The terms are not all positive. The first negative one is a(30) = -22690644647302814715858124800000. Conjecture: a(n) < 0 if and only if

A001221(n) is an odd number >= 3. - Robert Israel, May 15 2015

Links

Robert Israel, Table of n, a(n) for n = 1..411

Formula

MOBIUS transform of Harmonic Numbers is a(n)/n!. - Michael Somos, May 24 2015

a(n) = n! * Sum_{k=1..n} A191898(n,k)/k. - Mats Granvik, Jul 10 2016

Maple

for n from 1 to 50 do
  A[n]:= n! * (harmonic(n) - add(A[k]/k!, k = numtheory:-divisors(n) minus {n}))
od:
seq(A[n], n=1..50); # Robert Israel, May 15 2015

Mathematica

(*Recurrence:*)
Clear[t]; s = 1; nn = 20; t[1, 1] = 1;
t[n_, k_] :=
t[n, k] =
  If[k == 1, HarmonicNumber[n, s] - Sum[t[n, k + i], {i, 1, n - 1}],
   If[Mod[n, k] == 0, t[n/k, 1], 0], 0]; Table[t[n, 1]*n!, {n, 1, nn}]
(* Mats Granvik, May 14 2015 *)

Prog

(PARI) {a(n) = if( n<1, 0, n! * sumdiv(n, d, moebius(n/d) * sum(k=1, d, 1/k)))}; /* Michael Somos, May 24 2015 */

Crossrefs

Cf. A191898.

Sequence in context: A070135 A317037 A317691 * A027832 A128946 A156219

Adjacent sequences: A067854 A067855 A067856 * A067858 A067859 A067860

Keyword

sign

Author

Leroy Quet, Feb 15 2002

Status

approved