A130799 Triangle read by rows in which row n (n>=3) list the anti-divisors of n.

2, 3, 2, 3, 4, 2, 3, 5, 3, 5, 2, 6, 3, 4, 7, 2, 3, 7, 5, 8, 2, 3, 5, 9, 3, 4, 9, 2, 6, 10, 3, 11, 2, 3, 5, 7, 11, 4, 5, 7, 12, 2, 3, 13, 3, 8, 13, 2, 6, 14, 3, 4, 5, 9, 15, 2, 3, 5, 9, 15, 7, 16, 2, 3, 7, 10, 17, 3, 4, 17, 2, 5, 6, 11, 18, 3, 5, 8, 11, 19, 2, 3, 19, 4, 12, 20, 2, 3, 7

Offset: 3

Author

Diana L. Mecum, Jul 17 2007

Keywords

• nonn
• tabf

Comments

A066272 gives the number of terms in each row.

See A066272 for definition of anti-divisor.

2n-1 and 2n+1 are twin primes (that is, n is in A040040) iff n has no odd anti-divisors. For example, because n=15 has no odd anti-divisors, 29 and 31 are twin primes. - Jon Perry, Sep 12 2012

Row n is all the numbers which are: (a) 2n divided by its odd divisors (except 1), and (b) the divisors of 2n-1 and 2n+1 (except 1, 2n+1 and 2n-1). For example, n=18: odd divisors of 36 are {3,9} and 36/{3,9} = {4,12}; divisors of 35 are {5,7} and divisors of 37 are null (37 is prime). Therefore row 18 is 4,5,7 and 12. See A066542 for further explanation. - Bob Selcoe, Feb 24 2014

Examples

			Anti-divisors of 3 through 20:
3: 2
4: 3
5: 2, 3
6: 4
7: 2, 3, 5
8: 3, 5
9: 2, 6
10: 3, 4, 7
11: 2, 3, 7
12: 5, 8
13: 2, 3, 5, 9
14: 3, 4, 9
15: 2, 6, 10
16: 3, 11
17: 2, 3, 5, 7, 11
18: 4, 5, 7, 12
19: 2, 3, 13
20: 3, 8, 13
		

Links

• T. D. Noe, Rows n=3..1000 of triangle, flattened
• Diana Mecum, Rows 3 through 500

Programs

Mathematica

f[n_] := Complement[ Sort@ Join[ Select[ Union@ Flatten@ Divisors[{2 n - 1, 2 n + 1}], OddQ@ # && # < n &], Select[ Divisors[2 n], EvenQ@ # && # < n &]], Divisors@ n]; Flatten@ Table[ f@n, {n, 3, 32}] (* Robert G. Wilson v, Jul 17 2007 *)
Table[Select[Range[2, n - 1], Abs[Mod[n, #] - #/2] < 1 &], {n, 3, 31}] // Flatten (* Michael De Vlieger, Jun 14 2016, after Harvey P. Dale at A066272 *)