A348296 Irregular table T(n, k), n > 0, k = 1..A000120(n), read by rows; the n-th contains, in ascending order, the distinct powers of 2 summing to n.

1, 2, 1, 2, 4, 1, 4, 2, 4, 1, 2, 4, 8, 1, 8, 2, 8, 1, 2, 8, 4, 8, 1, 4, 8, 2, 4, 8, 1, 2, 4, 8, 16, 1, 16, 2, 16, 1, 2, 16, 4, 16, 1, 4, 16, 2, 4, 16, 1, 2, 4, 16, 8, 16, 1, 8, 16, 2, 8, 16, 1, 2, 8, 16, 4, 8, 16, 1, 4, 8, 16, 2, 4, 8, 16, 1, 2, 4, 8, 16, 32

Offset: 1

Author

Rémy Sigrist, Jul 18 2022

Keywords

• nonn
• tabf
• base
• easy

Examples

			Triangle T(n, k) begins:
  n   n-th row
  --  ------------
   1  [1]
   2  [2]
   3  [1, 2]
   4  [4]
   5  [1, 4]
   6  [2, 4]
   7  [1, 2, 4]
   8  [8]
   9  [1, 8]
  10  [2, 8]
  11  [1, 2, 8]
  12  [4, 8]
  13  [1, 4, 8]
  14  [2, 4, 8]
  15  [1, 2, 4, 8]
		

Links

• Michael De Vlieger, Table of n, a(n) for n = 1..10870 (rows n = 1..2000, flattened)
• Index entries for sequences related to binary expansion of n

Crossrefs

Cf. A000079, A000120, A006519, A053644, A059867, A065620, A133457.

Programs

Mathematica

Array[DeleteCases[Union@ NumberExpand[#, 2], 0] &, 32] // Flatten (* Michael De Vlieger, Jul 19 2022 *)

PARI

row(n) = { my (r=vector(hammingweight(n))); for (k=1, #r, n -= r[k] = 2^valuation(n, 2)); return (r) }

Formula

T(n, k) = 2^A133457(n, k).

T(n, 1) = A006519(n).

T(n, A000120(n)) = A053644(n).

Sum_{k = 1..A000120(n)} T(n, k) = n.

Sum_{k = 1..A000120(n)} T(n, k) * (-1)^(k-1) = A065620(n).

Product_{k = 1..A000120(n)} T(n, k) = A059867(n).

T(2*n, k) = 2*T(n, k).