

A130328


Triangle of differences between powers of 2, read by rows.


5



1, 3, 2, 7, 6, 4, 15, 14, 12, 8, 31, 30, 28, 24, 16, 63, 62, 60, 56, 48, 32, 127, 126, 124, 120, 112, 96, 64, 255, 254, 252, 248, 240, 224, 192, 128, 511, 510, 508, 504, 496, 480, 448, 384, 256
(list;
table;
graph;
refs;
listen;
history;
text;
internal format)



OFFSET

0,2


COMMENTS

Column 0 contains the Mersenne numbers A000225.
An even perfect number (A000396) is found in the triangle by reference to its matching exponent for the Mersenne prime p (A000043) thus: go to row 2p  1 and then column p  1 (remembering that the first position is column 0).
Likewise divisors of multiply perfect numbers, if not the multiply perfect numbers themselves, can also be found in this triangle. (End)


LINKS



FORMULA

t(n, k) = 2^n  2^k, where n is the row number and k is the column number, running from 0 to n  1. (If k is allowed to reach n, then the triangle would have an extra diagonal filled with zeros)  Alonso del Arte, Mar 13 2008


EXAMPLE

First few rows of the triangle are;
1;
3, 2;
7, 6, 4;
15, 14, 12, 8;
31, 30, 28, 24, 16;
63, 62, 60, 56, 48, 32;
...
a(5, 2) = 28 because 2^5 = 32, 2^2 = 4 and 32  4 = 28.


MATHEMATICA

ColumnForm[Table[2^n  2^k, {n, 15}, {k, 0, n  1}], Center] (* Alonso del Arte, Mar 13 2008 *)


CROSSREFS



KEYWORD



AUTHOR



EXTENSIONS



STATUS

approved



