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

1,2

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

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).

Example

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]

Mathematica

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

Prog

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

Crossrefs

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

Sequence in context: A287691 A227926 A357121 * A357120 A261358 A261356

Adjacent sequences: A348293 A348294 A348295 * A348297 A348298 A348299

Keyword

nonn,tabf,base,easy

Author

Rémy Sigrist, Jul 18 2022

Status

approved