A133995 Irregular array read by rows: n-th row contains (in numerical order) the positive integers <= n which are neither divisors of n nor are coprime to n. A 0 is put into row n if there are no such integers.

0, 0, 0, 0, 0, 4, 0, 6, 6, 4, 6, 8, 0, 8, 9, 10, 0, 4, 6, 8, 10, 12, 6, 9, 10, 12, 6, 10, 12, 14, 0, 4, 8, 10, 12, 14, 15, 16, 0, 6, 8, 12, 14, 15, 16, 18, 6, 9, 12, 14, 15, 18, 4, 6, 8, 10, 12, 14, 16, 18, 20, 0, 9, 10, 14, 15, 16, 18, 20, 21, 22, 10, 15, 20, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 6, 12, 15, 18, 21, 24

Offset: 1

Author

Leroy Quet, Oct 01 2007

Keywords

• nonn
• tabf

Comments

Row n has length A264441(n).

The number of nonzero entries in row n is A045763(n).

Row n has a 0 if every positive integer <= n is coprime to n or divides n.

From Michael De Vlieger, Aug 19 2017: (Start)

When row n is not empty (and here represented by 0), the terms of row n are composite, since primes p < n must either divide or be coprime to n and the empty product 1 both divides and is coprime to all numbers. For the following, let prime p divide n and prime q be coprime to n.

Row n is empty for n < 8 except n = 6.

There are two distinct species of term m of row n. The first are nondivisor regular numbers g in A272618(n) that divide some integer power e > 1 of n. In other words, these numbers are products of primes p that also divide n and no primes q that are coprime to n, yet g itself does not divide n. Prime powers n = p^k cannot have numbers g in A272618(n) since they have only one distinct prime divisor p; all regular numbers g = p^e with 0 <= e <= k divide p^k. The smallest n = 6 for which there is a number in A272618. The number 4 is the smallest composite and is equal to n = 4 thus must divide it; 4 is coprime to 5. The number 4 is neither coprime to nor a divisor of 6.

The second are numbers h in A272619(n) that are products of at least one prime p that divides n and one prime q that is coprime to n.

The smallest n = 8 for which there is a number in A272619 is 8; the number 6 is the product of the smallest two distinct primes. 6 divides 6 and is coprime to 7. The number 6 is neither coprime to nor a divisor of the prime power 8; 4 divides 8 and does not appear in a(8).

There can be no other species since primes p <= n divide n and q < n are coprime to n, and products of primes q exclusive of any p are coprime to n.

As a consequence of these two species, rows 1 <= n <= 5 and n = 7 are empty and thus have 0 in row n.

(End)

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 contains the positive integers <= 12 which are in neither of these two lists: 8,9,10.
The irregular triangle T(n, k) begins:
n\k 1  2  3  4  5  6  7 ...
1:  0
2:  0
3:  0
4:  0
5:  0
6:  4
7:  0
8:  6
9:  6
10: 4  6  8
11: 0
12: 8  9 10
13: 0
14: 4  6  8 10 12
15: 6  9 10 12
16: 6 10 12 14
17: 0
18: 4  8 10 12 14 15 16
19: 0
20: 6  8 12 14 15 16 18
... formatted by _Wolfdieter Lang_, Jan 16 2016
		

Links

• Robert Israel, Table of n, a(n) for n = 1..10014 (rows 1 to 237, flattened)

Crossrefs

Cf. A045763, A133994, A264441, A272618, A272619.

Programs

Maple

row:= proc(n) local r;
   r:= remove(t -> member(igcd(t, n), [1, t]), [$1..n]):
   if r = [] then 0 else op(r) fi
end proc:
A:= [seq](row(n), n=1..30); # Robert Israel, Jan 19 2016

Mathematica

Table[Select[Range@ n, Nor[Divisible[n, #], CoprimeQ[n, #]] &] /. {} -> {0}, {n, 27}] // Flatten (* Michael De Vlieger, Aug 19 2017 *)

Formula

a(n) = union(A272618(n), A272619(n)). - Michael De Vlieger, Aug 19 2017

Extensions

More terms from Alvin Hoover Belt, Jan 21 2008

Edited by Wolfdieter Lang, Jan 16 2016