A390279 Triangle read by rows: T(n, k) = Sum_{d | gcd(n, k)} d*Möbius(n/d).

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, -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, 1, -1, -4, -1, 1, -1, 1, 4, 10, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, 10

Offset

0,7

Links

Paolo Xausa, Table of n, a(n) for n = 0..11475 (rows 0..150 of triangle, flattened).

Formula

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

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

Example

Triangle starts:
  [0]  0;
  [1]  1,  1;
  [2]  1, -1,  1;
  [3]  2, -1, -1,  2;
  [4]  2,  0, -2,  0,  2;
  [5]  4, -1, -1, -1, -1,  4;
  [6]  2,  1, -1, -2, -1,  1,  2;
  [7]  6, -1, -1, -1, -1, -1, -1, 6;
  [8]  4,  0,  0,  0, -4,  0,  0, 0, 4;
  [9]  6,  0,  0, -3,  0,  0, -3, 0, 0, 6;

Maple

with(numtheory):
T := (n, k) -> local d; add(d*mobius(n/d), d in divisors(igcd(n, k))):
seq(seq(T(n, k), k = 0..n), n = 0..11);

Mathematica

A390279[n_, k_] := If[n == 0, 0, DivisorSum[GCD[n, k], #*MoebiusMu[n/#] &]];
Table[A390279[n, k], {n, 0, 15}, {k, 0, n}] (* Paolo Xausa, Oct 31 2025 *)

Crossrefs

Variant: A054533.

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

Sequence in context: A225803 A357458 A349277 * A382817 A307014 A240871

Adjacent sequences: A390276 A390277 A390278 * A390280 A390281 A390282

Keyword

sign,tabl

Author

Peter Luschny, Oct 31 2025

Status

approved