OFFSET
1,6
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)
LINKS
Robert Israel, Table of n, a(n) for n = 1..10014 (rows 1 to 237, flattened)
FORMULA
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 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
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 *)
CROSSREFS
KEYWORD
nonn,tabf
AUTHOR
Leroy Quet, Oct 01 2007
EXTENSIONS
More terms from Alvin Hoover Belt, Jan 21 2008
Edited by Wolfdieter Lang, Jan 16 2016
STATUS
approved