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A067857
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Sum_{k|n} a(k)/k! = Sum_{j=1 to n} 1/j, sum on left is over positive divisors k of n.
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1
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1, 1, 5, 14, 154, 84, 8028, 25584, 361296, 528480, 80627040, 33471360, 13575738240, 13835646720, 263577888000, 13869128448000, 867718162483200, 316745643110400, 309920046408806400, 207862451693568000
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OFFSET
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1,3
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COMMENTS
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The terms are not all positive. The first negative one is a(30) = -22690644647302814715858124800000. Conjecture: a(n) < 0 if and only if
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LINKS
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FORMULA
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MOBIUS transform of Harmonic Numbers is a(n)/n!. - Michael Somos, May 24 2015
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MAPLE
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for n from 1 to 50 do
A[n]:= n! * (harmonic(n) - add(A[k]/k!, k = numtheory:-divisors(n) minus {n}))
od:
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MATHEMATICA
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(*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}]
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PROG
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(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 */
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CROSSREFS
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KEYWORD
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sign
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AUTHOR
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STATUS
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approved
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