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

1,3

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

Links

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

Example

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

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 *)

Crossrefs

Cf. A073757, A133995.

Sequence in context: A243732 A194905 A243730 * A066041 A194965 A243712

Adjacent sequences: A133991 A133992 A133993 * A133995 A133996 A133997

Keyword

nonn,tabf

Author

Leroy Quet, Oct 01 2007

Status

approved