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

0,2

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)

Links

Table of n, a(n) for n=0..44.

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

Cf. A130321, A059268, A000337, A130329.

Sequence in context: A072764 A364832 A349890 * A228993 A083569 A071574

Adjacent sequences: A130325 A130326 A130327 * A130329 A130330 A130331

Keyword

nonn,tabl

Author

Gary W. Adamson, May 24 2007

Extensions

Better definition from Alonso del Arte, Mar 13 2008

Status

approved