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Triangle read by rows: T(n,k) = phi(n/k)*A023900(k) if k divides n, T(n,k) = 0 otherwise (n >= 1, 1 <= k <= n).
1

%I #21 Oct 14 2023 14:05:17

%S 1,1,-1,2,0,-2,2,-1,0,-1,4,0,0,0,-4,2,-2,-2,0,0,2,6,0,0,0,0,0,-6,4,-2,

%T 0,-1,0,0,0,-1,6,0,-4,0,0,0,0,0,-2,4,-4,0,0,-4,0,0,0,0,4,10,0,0,0,0,0,

%U 0,0,0,0,-10,4,-2,-4,-2,0,2,0,0,0,0,0,2

%N Triangle read by rows: T(n,k) = phi(n/k)*A023900(k) if k divides n, T(n,k) = 0 otherwise (n >= 1, 1 <= k <= n).

%C Sum_{k=1..n} T(n,k) = A063524(n).

%F T(n,k) = phi(n/k)*A023900(k) if k divides n, T(n,k) = 0 otherwise (n >= 1, 1 <= k <= n).

%e {

%e {1}, = 1

%e {1, -1}, = 0

%e {2, 0, -2}, = 0

%e {2, -1, 0, -1}, = 0

%e {4, 0, 0, 0, -4}, = 0

%e {2, -2, -2, 0, 0, 2}, = 0

%e {6, 0, 0, 0, 0, 0, -6}, = 0

%e {4, -2, 0, -1, 0, 0, 0, -1}, = 0

%e {6, 0, -4, 0, 0, 0, 0, 0, -2}, = 0

%e {4, -4, 0, 0, -4, 0, 0, 0, 0, 4}, = 0

%e {10, 0, 0, 0, 0, 0, 0, 0, 0, 0, -10}, = 0

%e {4, -2, -4, -2, 0, 2, 0, 0, 0, 0, 0, 2} = 0

%e }

%t nn = 12; g[n_] := DivisorSum[n, MoebiusMu[#] # &]; Flatten[Table[Table[If[Mod[n, k] == 0, EulerPhi[n/k]*g[k], 0], {k, 1, n}], {n, 1, nn}]]

%Y Cf. A000010, A023900, A366445, A054524, A054525, A063524, A054522, A054523, A129691, A127649.

%K sign,tabl

%O 1,4

%A _Mats Granvik_, Oct 12 2023