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A068337 a(n) = n!*Sum_{k=1..n} mu(k)/k, where mu(k) is the Möbius function.

%I A068337 #28 Nov 03 2023 17:02:02
%S A068337 1,1,1,4,-4,96,-48,-384,-3456,328320,-17280,-207360,-481697280,
%T A068337 -516741120,79427174400,1270834790400,681401548800,12265227878400,
%U A068337 -6169334376038400,-123386687520768000,-158218429759488000,47610136717000704000
%N A068337 a(n) = n!*Sum_{k=1..n} mu(k)/k, where mu(k) is the Möbius function.
%H A068337 Amiram Eldar, <a href="/A068337/b068337.txt">Table of n, a(n) for n = 1..450</a>
%H A068337 Friedrich Roesler, <a href="https://doi.org/10.1016/0024-3795(86)90255-7">Riemann's hypothesis as an eigenvalue problem</a>, Linear Algebra and its Applications, Vol. 81 (1986), pp. 153-198.
%H A068337 Friedrich Roesler, <a href="https://doi.org/10.1016/0024-3795(87)90250-3">Riemann's hypothesis as an eigenvalue problem. II</a>, Linear Algebra and its Applications, Vol. 92 (1987), pp. 45-73.
%F A068337 a(n) = (-1)^(n-1)*{determinant of the n X n matrix m(i,j) = i+(j (mod i))} - _Benoit Cloitre_, May 28 2004
%F A068337 From _Amiram Eldar_, Oct 22 2020: (Start)
%F A068337 a(n) = A000142(n)*A070888(n)/A070889(n).
%F A068337 a(n) ~ O(n! * n^(-1/2 + eps)), for every eps>0, if and only if Riemann's hypothesis is true (Roesler, 1986). (End)
%t A068337 n = 25; Accumulate[Table[MoebiusMu[k]/k, {k, 1, n}]] * Range[n]! (* _Amiram Eldar_, Oct 22 2020 *)
%o A068337 (Python)
%o A068337 from math import factorial
%o A068337 from functools import lru_cache
%o A068337 from sympy import harmonic
%o A068337 @lru_cache(maxsize=None)
%o A068337 def f(n):
%o A068337     if n <= 1:
%o A068337         return 1
%o A068337     c, j = 1, 2
%o A068337     k1 = n//j
%o A068337     while k1 > 1:
%o A068337         j2 = n//k1 + 1
%o A068337         c += (harmonic(j-1)-harmonic(j2-1))*f(k1)
%o A068337         j, k1 = j2, n//j2
%o A068337     return c+harmonic(j-1)-harmonic(n)
%o A068337 def A068337(n): return factorial(n)*f(n) # _Chai Wah Wu_, Nov 03 2023
%Y A068337 Cf. A000142, A008683, A070888, A070889.
%K A068337 sign
%O A068337 1,4
%A A068337 _Leroy Quet_, Feb 27 2002