A233757 Triangle read by rows: T(n,k) = (2^n-1)*2^(k-1), for n >= 1 and 1<=k<=n.

1, 3, 6, 7, 14, 28, 15, 30, 60, 120, 31, 62, 124, 248, 496, 63, 126, 252, 504, 1008, 2016, 127, 254, 508, 1016, 2032, 4064, 8128, 255, 510, 1020, 2040, 4080, 8160, 16320, 32640, 511, 1022, 2044, 4088, 8176, 16352, 32704, 65408, 130816, 1023, 2046, 4092

Offset: 1

Author

Omar E. Pol, Jan 12 2014

Keywords

• nonn
• tabl

Comments

Column 1 gives the positive terms of A000225.

Leading diagonal gives the positive terms of A006516.

The sum of row n is T(n,1)^2 = A000225(n)^2, hence row sums give A060867.

If n = A000043(m) then T(n,1) = A000668(m) and row n lists last n divisors of m-th even perfect number, which are also the divisors that are multiples of m-th Mersenne prime, for m >= 1.

If n = A000043(m) then T(n,n) = A000396(m), assuming there are no odd perfect numbers, for m >= 1.

Examples

			Triangle begins:
1;
3, 6;
7, 14, 28;
15, 30, 60, 120;
31, 62, 124, 248, 496;
63, 126, 252, 504, 1008, 2016;
127, 254, 508, 1016, 2032, 4064, 8128;
255, 510, 1020, 2040, 4080, 8160, 16320, 32640;
511, 1022, 2044, 4088, 8176, 16352, 32704, 65408, 130816;
...
		

Links

• Harvey P. Dale, Table of n, a(n) for n = 1..5000

Crossrefs

Cf. A000043, A000079, A000225, A000396, A000668, A006516, A018254, A018487, A059268, A060867, A133024, A133025, A135652-A135655, A139247.

Programs

Mathematica

Table[(2^n-1)2^(k-1),{n,10},{k,n}]//Flatten (* Harvey P. Dale, Oct 10 2018 *)

Formula

T(n,k) = A000225(n)*A000079(k-1), n >= 1, 1<=k<=n.