A300288 Irregular triangle read by rows: row n lists the numbers k such that -n <= k <= n and gcd(k, 2*n+1) != 1.

0, 0, 0, -3, 0, 3, 0, 0, -6, -5, -3, 0, 3, 5, 6, 0, 0, -9, -7, -6, -3, 0, 3, 6, 7, 9, 0, -10, -5, 0, 5, 10, -12, -9, -6, -3, 0, 3, 6, 9, 12, 0, 0, -15, -12, -11, -9, -6, -3, 0, 3, 6, 9, 11, 12, 15, -15, -14, -10, -7, -5, 0, 5, 7, 10, 14, 15, 0, -18, -15, -13, -12, -9, -6, -3, 0, 3, 6, 9, 12, 13, 15, 18, 0

Offset: 1

Author

Joerg Arndt, Mar 02 2018

Keywords

• sign
• tabf

Comments

Row n contains 2*n+1 - phi(2*n+1) = A053193(n) terms. Row n has just one term (namely 0) if 2*n+1 is prime.

Examples

			Triangle starts:
[01]:  0,
[02]:  0,
[03]:  0,
[04]: -3, 0, 3,
[05]:  0,
[06]:  0,
[07]: -6, -5, -3, 0, 3, 5, 6,
[08]:  0,
[09]:  0,
[10]: -9, -7, -6, -3, 0, 3, 6, 7, 9,
[11]:  0,
[12]: -10, -5, 0, 5, 10,
[13]: -12, -9, -6, -3, 0, 3, 6, 9, 12,
[14]:  0,
[15]:  0,
[16]: -15, -12, -11, -9, -6, -3, 0, 3, 6, 9, 11, 12, 15,
[17]: -15, -14, -10, -7, -5, 0, 5, 7, 10, 14, 15,
[18]:  0,
[19]: -18, -15, -13, -12, -9, -6, -3, 0, 3, 6, 9, 12, 13, 15, 18,
[20]:  0,
...
		

Links

• Paolo Xausa, Table of n, a(n) for n = 1..11968 (rows 1..250 of the triangle, flattened)

Crossrefs

Cf. A053193, A299714.

Programs

Mathematica

A300288row[n_]:=With[{q=2n+1},If[PrimeQ[q],{0},Select[Range[-n, n],GCD[#, q]!=1&]]];Array[A300288row,20] (* Paolo Xausa, Nov 10 2023 *)

PARI

is(n, k)= ( gcd(k, 2*n+1)!=1 );
for (n=1, 33, for (k=-n, +n, if (is(n, k), print1(k, ", ") ); ); );

Formula

T(n,k) = A299714(n,k) - n.