A335404 Numbers k such that the k-th composition in standard order (A066099) has the same product as sum.

1, 2, 4, 8, 10, 16, 32, 37, 38, 41, 44, 50, 52, 64, 128, 139, 141, 142, 163, 171, 173, 174, 177, 181, 182, 184, 186, 197, 198, 209, 213, 214, 216, 218, 226, 232, 234, 256, 295, 307, 313, 316, 403, 409, 412, 457, 460, 484, 512, 535, 539, 541, 542, 647, 707, 737

Offset

1,2

Comments

The k-th composition in standard order (graded reverse-lexicographic, A066099) is obtained by taking the set of positions of 1's in the reversed binary expansion of k, prepending 0, taking first differences, and reversing again. This gives a bijective correspondence between nonnegative integers and integer compositions.

Links

Table of n, a(n) for n=1..56.

Gus Wiseman, Statistics, classes, and transformations of standard compositions

Formula

A124758(a(n)) = A070939(a(n)).

Example

The sequence together with the corresponding compositions begins:
    1: (1)
    2: (2)
    4: (3)
    8: (4)
   10: (2,2)
   16: (5)
   32: (6)
   37: (3,2,1)
   38: (3,1,2)
   41: (2,3,1)
   44: (2,1,3)
   50: (1,3,2)
   52: (1,2,3)
   64: (7)
  128: (8)
  139: (4,2,1,1)
  141: (4,1,2,1)
  142: (4,1,1,2)
  163: (2,4,1,1)
  171: (2,2,2,1,1)

Mathematica

stc[n_]:=Differences[Prepend[Join@@Position[Reverse[IntegerDigits[n, 2]], 1], 0]]//Reverse;
Select[Range[0, 100], Times@@stc[#]==Plus@@stc[#]&]

Crossrefs

The lengths of standard compositions are given by A000120.

Sum of binary indices is A029931.

Sum of prime indices is A056239.

Sum of standard compositions is A070939.

Product of standard compositions is A124758.

Taking GCD instead of product gives A131577.

The version for prime indices is A301987.

The version for prime indices of nonprime numbers is A301988.

These compositions are counted by A335405.

Cf. A001055, A003963, A066099, A096111, A124767, A228351, A233249, A272919, A333219, A333220, A331579.

Sequence in context: A337666 A291165 A083655 * A353431 A353696 A271816

Adjacent sequences: A335401 A335402 A335403 * A335405 A335406 A335407

Keyword

nonn

Author

Gus Wiseman, Jun 06 2020

Status

approved