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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, 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
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
KEYWORD
sign
AUTHOR
Leroy Quet, Feb 15 2002
STATUS
approved