| Here are some multiplication table fragments. Only half of the
table is shown (because multiplication is symmetric), and an entry
in any row that would duplicate an entry in a previous row is
left blank.
*| 1| 2| 3| 4| 5| 6| 7| 8| 9| 10| 11| 12| 13| 14| 15| 16|
--+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+
1| 1|
2| 2| 4|
3| 3| 6| 9|
4| | 8| 12| 16|
5| 5| 10| 15| 20| 25|
6| | | 18| 24| 30| 36|
7| 7| 14| 21| 28| 35| 42| 49|
8| | | | 32| 40| 48| 56| 64|
9| | | 27| 36| 45| 54| 63| 72| 81|
10| | | | | 50| 60| 70| 80| 90|100|
11| 11| 22| 33| 44| 55| 66| 77| 88| 99|110|121|
12| | | | | | | 84| 96|108|120|132|144|
13| 13| 26| 39| 52| 65| 78| 91|104|117|130|143|156|169|
14| | | | | | | 98|112|126|140|154|169|182|196|
15| | | | | | | | |135|150|165|180|195|210|225|
16| | | | | | | |128|144|160!176|192|208|224|240|256|
--+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+
*| 1| 2| 3| 4| 5| 6| 7| 8| 9| 10| 11| 12| 13| 14| 15| 16|
Notice that each row contains some number of blank entries
followed by some number of non-blank entries. Prime numbers
have no blank entries (no multiple of 7 can occur until row 7).
For the first 9 rows, the number of blank entries in row n in
the number of divisors of n excluding 1 and n, but that nice
rule fell apart for 10, 12, 14, 15, and 16.
Zero was left out of the table above, but the resulting formula
might appear nicer if zero is included. Here are some summary
results. I have gone nowhere with this.
B.J.
| | new | 1-based | 0-based |
n | blanks | numbers | total | total |
----+--------+---------+---------+---------+
0 | | 1 | - | 1 |
1 | 0 | 1 | 1 | 2 |
2 | 0 | 2 | 3 | 4 |
3 | 0 | 3 | 6 | 7 |
4 | 1 | 3 | 9 | 10 |
5 | 0 | 5 | 14 | 15 |
6 | 2 | 4 | 18 | 19 |
7 | 0 | 7 | 25 | 26 |
8 | 3 | 5 | 30 | 31 |
9 | 2 | 7 | 37 | 38 |
10 | 4 | 6 | 43 | 44 |
11 | 0 | 11 | 54 | 55 |
12 | 6 | 6 | 60 | 61 |
13 | 0 | 13 | 73 | 74 |
14 | 6 | 8 | 81 | 82 |
15 | 8 | 7 | 88 | 89 |
16 | 7 | 9 | 97 | 98 |
17 | 0 | 17 | 114 | 115 |
|