A130328 Triangle of differences between powers of 2, read by rows.

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

Offset: 0

Author

Gary W. Adamson, May 24 2007

Keywords

• nonn
• tabl

Comments

A130321 * A059268 as infinite lower triangular matrices.

Row sums = A000337: (1, 5, 17, 49, 129, 321, ...). A130329 = A059268 * A130321.

From Alonso del Arte, Mar 13 2008: (Start)

Column 0 contains the Mersenne numbers A000225.

Column 1 is A000918.

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)

Examples

			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.
		

Crossrefs

Cf. A130321, A059268, A000337, A130329.

Programs

Mathematica

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

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

Extensions

Better definition from Alonso del Arte, Mar 13 2008