A390279 - OEIS

%I #12 Oct 31 2025 12:16:23

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

%T -1,-1,-1,-1,-1,-1,6,4,0,0,0,-4,0,0,0,4,6,0,0,-3,0,0,-3,0,0,6,4,1,-1,

%U 1,-1,-4,-1,1,-1,1,4,10,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,10

%N Triangle read by rows: T(n, k) = Sum_{d | gcd(n, k)} d*Möbius(n/d).

%H Paolo Xausa, <a href="/A390279/b390279.txt">Table of n, a(n) for n = 0..11475</a> (rows 0..150 of triangle, flattened).

%F Sum_{k=0..4*n} (-1)^k*T(4*n, k) = 2*EulerPhi(2*n).

%F -T(2*n, n) = EulerPhi(2*n) = A062570(n).

%e Triangle starts:

%e   [0]  0;

%e   [1]  1,  1;

%e   [2]  1, -1,  1;

%e   [3]  2, -1, -1,  2;

%e   [4]  2,  0, -2,  0,  2;

%e   [5]  4, -1, -1, -1, -1,  4;

%e   [6]  2,  1, -1, -2, -1,  1,  2;

%e   [7]  6, -1, -1, -1, -1, -1, -1, 6;

%e   [8]  4,  0,  0,  0, -4,  0,  0, 0, 4;

%e   [9]  6,  0,  0, -3,  0,  0, -3, 0, 0, 6;

%p with(numtheory):

%p T := (n, k) -> local d; add(d*mobius(n/d), d in divisors(igcd(n, k))):

%p seq(seq(T(n, k), k = 0..n), n = 0..11);

%t A390279[n_, k_] := If[n == 0, 0, DivisorSum[GCD[n, k], #*MoebiusMu[n/#] &]];

%t Table[A390279[n, k], {n, 0, 15}, {k, 0, n}] (* _Paolo Xausa_, Oct 31 2025 *)

%Y Variant: A054533.

%Y Cf. A384710 (row sums), A000010, A008683, A062570.

%K sign,tabl

%O 0,7

%A _Peter Luschny_, Oct 31 2025