A007431 -id:A007431 - cp's OEIS frontend

A127505 Triangle T(n,k) = mobius(n/k)*phi(k) if k|n, otherwise T(n,k)=0; 1<=k<=n.

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

Offset: 1

Gary W. Adamson, Jan 17 2007

			First few rows of the triangle are:
1;
-1, 1;
-1, 0, 2;
0, -1, 0, 2;
-1, 0, 0, 0, 4;
1, -1, -2, 0, 0, 2;
...

Cf. A051731, A000010 (diagonal n=k), A007431 (row sums), A008683 (column k=1).

T(n,k) = sum_{j=k..n} A054525(n,j)*A130207(j,k), 1<=k<=n.

A369700 Möbius transform of reduced totient function (A002322).

Original entry on oeis.org

1, 0, 1, 1, 3, 0, 5, 0, 4, 0, 9, -1, 11, 0, -1, 2, 15, 0, 17, -1, -1, 0, 21, 0, 16, 0, 12, -1, 27, 0, 29, 4, -1, 0, 3, 0, 35, 0, -1, 0, 39, 0, 41, -1, 4, 0, 45, 0, 36, 0, -1, -1, 51, 0, 7, 0, -1, 0, 57, 1, 59, 0, -4, 8, -3, 0, 65, -1, -1, 0

Offset: 1

Miles Englezou, Jan 29 2024

sign

Since A002322(n) = A000010(n) for n = 1, 2, 4, and odd prime powers, a(n) = A007431(n) for the same values of n.

			a(8) = mu(1)*lambda(8) + mu(2)*lambda(4) + mu(4)*lambda(2) + mu(8)*lambda(1) = 0.

Cf. A008683, A002322, A000010, A007431.

a[n_] := DivisorSum[n, MoebiusMu[#] * CarmichaelLambda[n/#] &]; Array[a, 100] (* Amiram Eldar, Jan 29 2024 *)

a(n) = sumdiv(n, d, moebius(d)*lcm(znstar(n/d)[2]))

a(n) = Sum_{d|n} A008683(d) * A002322(n/d).

cp's OEIS Frontend

A127505 Triangle T(n,k) = mobius(n/k)*phi(k) if k|n, otherwise T(n,k)=0; 1<=k<=n.

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A369700 Möbius transform of reduced totient function (A002322).

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