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Positions of the Dirichlet inverse of the Euler totient function expansion of the partial sums of the Möbius inverse of the Harmonic numbers in the set of the sorted tuples that have upper and lower bounds -(k-1) and +(k-1).
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%I #13 May 06 2020 02:42:58

%S 1,1,3,11,197,664,25522,283333

%N Positions of the Dirichlet inverse of the Euler totient function expansion of the partial sums of the Möbius inverse of the Harmonic numbers in the set of the sorted tuples that have upper and lower bounds -(k-1) and +(k-1).

%C The sorting in the Mathematica program is lexicographic.

%F See program.

%e M={{1}} where the last entry is {1}.

%e A={{1}} where the last entry {1} in M is found at position 1 in the ordered set of tuples A, therefore a(1)=1.

%e M={{1}, {2, -1}} where the last entry is {2,-1}.

%e A={{2, -1}, {2, 0}, {2, 1}} where the last entry {2,-1} in M is found at position 1 in the ordered set of tuples A, therefore a(2)=1.

%e M={{1}, {2, -1}, {3, 0, -2}} where the last entry is {3, 0, -2}.

%e A={{3, -1, -2}, {3, -1, -1}, {3, 0, -2}, {3, -1, 0}, {3, 0, -1}, {3, 1, -2}, {3, -1, 1}, {3, 0, 0}, {3, 1, -1}, {3, -1, 2}, {3, 0, 1}, {3, 1, 0}, {3, 0, 2}, {3, 1, 1}, {3, 1, 2}} where the last entry {3, 0, -2} in M is found at position 3 in the ordered set of tuples A, therefore a(3)=3.

%t Monitor[Flatten[Table[

%t nnn = nnnn;

%t g1 = Table[

%t T = Tuples[

%t Table[Table[If[k == 1, nn, n], {n, -(k - 1), k - 1}], {k, 1,

%t nn}]];

%t b = Sort[

%t Table[{T[[n]], Total[T[[n]]/Range[Length[T[[n]]]]] - nn}, {n,

%t 1, Length[T]}], #1[[2]] < #2[[2]] &], {nn, 1, nnn}];

%t A = Table[b[[n]][[1]], {n, 1, Length[b]}];

%t Clear[T, n, k, b, a];

%t nn = nnn;

%t a[n_] := If[n < 1, 0, Sum[d MoebiusMu@d, {d, Divisors[n]}]];

%t M = Table[

%t Table[Sum[If[n >= k, a[GCD[n, k]], 0], {n, 1, m}], {k, 1,

%t m}], {m, 1, nn}];

%t Flatten[Position[A, M[[nnn]]]], {nnnn, 1, 8}]], nnnn]

%K nonn,more

%O 1,3

%A _Mats Granvik_, Feb 23 2020