A299714 Irregular triangle read by rows: row n contains numbers k such that 1<=k<=2*n+1 and gcd(n-k, 2*n+1) != 1.

1, 2, 3, 1, 4, 7, 5, 6, 1, 2, 4, 7, 10, 12, 13, 8, 9, 1, 3, 4, 7, 10, 13, 16, 17, 19, 11, 2, 7, 12, 17, 22, 1, 4, 7, 10, 13, 16, 19, 22, 25, 14, 15, 1, 4, 5, 7, 10, 13, 16, 19, 22, 25, 27, 28, 31, 2, 3, 7, 10, 12, 17, 22, 24, 27, 31, 32, 18, 1, 4, 6, 7, 10, 13, 16, 19, 22, 25, 28, 31, 32, 34, 37, 20, 21, 1, 2, 4, 7, 10, 12, 13

Offset: 1

Author

Joerg Arndt, Feb 17 2018

Keywords

• nonn
• tabf

Comments

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

If we extend each row [r,s, ..., z] to infinite length by appending [j*N*r,j*N*s, ..., j*N*z] for j>=1 and N=2*n+1 we obtain a relation on pairs of positive numbers (u,v). Write R(u,v) if v appears in row u (the relation is 1 on the pair (u,v)). Then R(u,u), R(u,v) ==> R(v, u), R(u,v) ==> R(u, v mod (2*u+1) ), and R(u,v) and v > u ==> R(u-(v-u), v-(v-u)) = R(2*u - v, u).

Examples

			Triangle starts:
[01]: [1]
[02]: [2]
[03]: [3]
[04]: [1, 4, 7]
[05]: [5]
[06]: [6]
[07]: [1, 2, 4, 7, 10, 12, 13]
[08]: [8]
[09]: [9]
[10]: [1, 3, 4, 7, 10, 13, 16, 17, 19]
[11]: [11]
[12]: [2, 7, 12, 17, 22]
[13]: [1, 4, 7, 10, 13, 16, 19, 22, 25]
[14]: [14]
[15]: [15]
[16]: [1, 4, 5, 7, 10, 13, 16, 19, 22, 25, 27, 28, 31]
[17]: [2, 3, 7, 10, 12, 17, 22, 24, 27, 31, 32]
[18]: [18]
...
		

Links

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

Crossrefs

Cf. A300288.

Programs

Maple

T:= n-> select(k-> igcd(n-k, 2*n+1)<>1, [$1..2*n+1])[]:
seq(T(n), n=1..25);  # Alois P. Heinz, Mar 09 2018

Mathematica

A299714row[n_]:=With[{q=2n+1},If[PrimeQ[q],{n},Select[Range[q],GCD[n-#,q]!=1&]]];Array[A299714row,20] (* Paolo Xausa, Nov 10 2023 *)

PARI

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

Formula

T(n,k) = A300288(n,k) - n.