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%I #37 Jul 11 2016 14:57:42
%S 1,1,5,14,154,84,8028,25584,361296,528480,80627040,33471360,
%T 13575738240,13835646720,263577888000,13869128448000,867718162483200,
%U 316745643110400,309920046408806400,207862451693568000
%N Sum_{k|n} a(k)/k! = Sum_{j=1 to n} 1/j, sum on left is over positive divisors k of n.
%C The terms are not all positive. The first negative one is a(30) = -22690644647302814715858124800000. Conjecture: a(n) < 0 if and only if
%C A001221(n) is an odd number >= 3. - _Robert Israel_, May 15 2015
%H Robert Israel, <a href="/A067857/b067857.txt">Table of n, a(n) for n = 1..411</a>
%F MOBIUS transform of Harmonic Numbers is a(n)/n!. - _Michael Somos_, May 24 2015
%F a(n) = n! * Sum_{k=1..n} A191898(n,k)/k. - _Mats Granvik_, Jul 10 2016
%p for n from 1 to 50 do
%p A[n]:= n! * (harmonic(n) - add(A[k]/k!, k = numtheory:-divisors(n) minus {n}))
%p od:
%p seq(A[n],n=1..50); # _Robert Israel_, May 15 2015
%t (*Recurrence:*)
%t Clear[t]; s = 1; nn = 20; t[1, 1] = 1;
%t t[n_, k_] :=
%t t[n, k] =
%t If[k == 1, HarmonicNumber[n, s] - Sum[t[n, k + i], {i, 1, n - 1}],
%t If[Mod[n, k] == 0, t[n/k, 1], 0], 0]; Table[t[n, 1]*n!, {n, 1, nn}]
%t (* _Mats Granvik_, May 14 2015 *)
%o (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 */
%Y Cf. A191898.
%K sign
%O 1,3
%A _Leroy Quet_, Feb 15 2002
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