A130799 - OEIS

%I #37 Jun 15 2016 09:05:26

%S 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,

%T 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,

%U 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

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

%C A066272 gives the number of terms in each row.

%C See A066272 for definition of anti-divisor.

%C 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

%C 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

%H T. D. Noe, <a href="/A130799/b130799.txt">Rows n=3..1000 of triangle, flattened</a>

%H Diana Mecum, <a href="/A130799/a130799.txt">Rows 3 through 500</a>

%e Anti-divisors of 3 through 20:

%e 3: 2

%e 4: 3

%e 5: 2, 3

%e 6: 4

%e 7: 2, 3, 5

%e 8: 3, 5

%e 9: 2, 6

%e 10: 3, 4, 7

%e 11: 2, 3, 7

%e 12: 5, 8

%e 13: 2, 3, 5, 9

%e 14: 3, 4, 9

%e 15: 2, 6, 10

%e 16: 3, 11

%e 17: 2, 3, 5, 7, 11

%e 18: 4, 5, 7, 12

%e 19: 2, 3, 13

%e 20: 3, 8, 13

%t 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 *)

%t 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 *)

%K nonn,tabf

%O 3,1

%A _Diana L. Mecum_, Jul 17 2007