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%I #18 Dec 17 2019 05:36:39
%S 1,1,4,27,64,3125,288,823543,147456,4251528,460800,285311670611,
%T 111974400,302875106592253,3251404800,13436928000,106542032486400,
%U 827240261886336764177,1053455155200000,1978419655660313589123979,102395841085440000
%N Derived from von Mangoldt matrix sequence.
%C Since the logarithm of n is given by the limit of Zeta(s)*Sum_{k=1..n} ((1 - (If k mod n = 0 then n else 0))/k^(s - 1)) as s -> 1, it is natural to ask what the von Mangoldt function variant might look like starting from the table A191898, instead of table A167407. - _Mats Granvik_, Nov 11 2013
%F a(prime(n)) = A000312(prime(n)).
%t Clear[nn, t, n, k, i, s]; nn = 20; t[n_, 1] = 1; t[1, k_] = 1; t[n_, k_] := t[n, k] = If[n >= k, -Sum[t[n - i, k], {i, 1, k - 1}], -Sum[t[k - i, n], {i, 1, n - 1}]]; Exp[Table[Limit[Zeta[s]*Sum[If[n == 1, 0, t[n, k]]/k^(s - 1), {k, 1, n}], s -> 1], {n, 0, nn}]]*(Range[nn + 1] - 1)!
%Y Cf. A000312, A036505, A177885.
%K nonn
%O 0,3
%A _Mats Granvik_, Nov 02 2013
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