A133994 Irregular array read by rows: n-th row contains (in numerical order) both the positive integers <= n that are divisors of n and those that are coprime to n.

1, 1, 2, 1, 2, 3, 1, 2, 3, 4, 1, 2, 3, 4, 5, 1, 2, 3, 5, 6, 1, 2, 3, 4, 5, 6, 7, 1, 2, 3, 4, 5, 7, 8, 1, 2, 3, 4, 5, 7, 8, 9, 1, 2, 3, 5, 7, 9, 10, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 1, 2, 3, 4, 5, 6, 7, 11, 12, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 1, 2, 3, 5, 7, 9, 11, 13, 14

Offset: 1

Author

Leroy Quet, Oct 01 2007

Keywords

• nonn
• tabf

Comments

Row n contains A073757(n) terms.

The number 1 would appear twice for each n >= 1 if one takes the union of the divisor list of n and the list of the smallest positive reduced residue system modulo n. - Wolfdieter Lang, Jan 16 2016

Examples

			The divisors of 12 are: 1,2,3,4,6,12. The positive integers which are <= 12 and are coprime to 12 are: 1,5,7,11. So row 12 is the union of these two sets: 1,2,3,4,5,6,7,11,12.
The irregular triangle T(n, k) starts:
n\k 1 2 3 4 5 6  7  8  9 10 11 12 13 14 15 16 17
1:  1
2:  1 2
3:  1 2 3
4:  1 2 3 4
5:  1 2 3 4 5
6:  1 2 3 5 6
7:  1 2 3 4 5 6  7
8:  1 2 3 4 5 7  8
9:  1 2 3 4 5 7  8  9
10: 1 2 3 5 7 9 10
11: 1 2 3 4 5 6  7  8  9 10 11
12: 1 2 3 4 5 6  7 11 12
13: 1 2 3 4 5 6  7  8  9 10 11 12 13
14: 1 2 3 5 7 9 11 13 14
15: 1 2 3 4 5 7  8 11 13 14 15
16: 1 2 3 4 5 7  8  9 11 13 15 16
17: 1 2 3 4 5 6  7  8  9 10 11 12 13 14 15 16 17
18: 1 2 3 5 6 7  9 11 13 17 18 19
... Formatted by _Wolfdieter Lang_, Jan 16 2016
		

Links

• Robert Israel, Table of n, a(n) for n = 1..10003 (rows 1 to 174, flattened)

Crossrefs

Cf. A073757, A133995.

Programs

Maple

row:= n -> op(select(t -> member(igcd(t,n), [1,t]), [$1..n])):
seq(row(n), n=1..30); # Robert Israel, Jan 18 2016

Mathematica

row[n_] := Divisors[n] ~Union~ Select[Range[n], CoprimeQ[n, #]&]; Array[ row, 15] // Flatten (* Jean-François Alcover, Jan 18 2016 *)