- 8,128 is the 127th triangular number, the 64th hexagonal number, the eighth 292-gonal number, and the fourth 1356-gonal number.
I just noticed a pattern to this. 8128 is the xth 3-, 6-, 292-, and 1356-gonal number, with x taking the values 4, 8, 64, 127. I have a feeling the above ought to say 8128 is the 128th triangular number. The sequence 4, 8, 64, 127 yields no results from Sloane's, but 4, 8, 64, 128 does. PrimeFan 22:42, 1 Oct 2004 (UTC)
- One property that 8128 doesn't share with all the other perfect numbers is that it is the sum of every fourth entry of a row in Pascal's triangle.
Removed, because I can't make head or tail of it. I've tried summing every fourth number in each row of Pascal's triangle, and none come to 8128. Does this mean something else? sjorford →•← 09:40, 9 Jun 2005 (UTC)
- An OEIS search for ",8128, AND pascal" gives five results, including A038504 Sum of every 4th entry of row n in Pascal's triangle, starting at "n choose 1" and A038505 Sum of every 4th entry of row n in Pascal's triangle, starting at "n choose 2". It seems to me though that this requires further investigation and clarification before restoration. Anton Mravcek 19:15, 9 Jun 2005 (UTC)
- Right, I've worked it out from the OEIS entry - take the second diagonal of PT (the top right-bottom left diagonal that goes 1,2,3,4,5,6,7...), and on each row of the triangle sum every fourth number beginning with that diagonal. This gives 8128 on row 16:
1 15 105 455 1365 3003 5005 6435 6435 5005 3003 1365 455 105 15 1
...15 + 3003 + 5005 + 105 = 8128.
Whether this is interesting enough to be in this article, I'm not sure - we can't include every time 8128 appears in OEIS! Perhaps it's saying it's the only perfect number with this property, but I'm not certain. sjorford →•← 20:43, 9 Jun 2005 (UTC)