A096352 Triangle read by rows: each row represents all possible values for the size of the subset S{n - x} of {2^n...2^(n+1) - 1}, where S{n - x} represents all the members of that set with n - x factors.

2, 4, 5, 2, 4, 6, 7, 8, 5, 12, 17, 20, 21, 22, 7, 20, 30, 37, 41, 44, 46, 47, 13, 40, 65, 81, 91, 96, 99, 101, 102, 103, 23, 75, 131, 173, 199, 215, 224, 229, 232, 233, 234, 43, 147, 257, 344, 403, 439, 461, 473, 482, 487, 490, 492, 493

Offset: 1

Author

Andrew S. Plewe, Jun 29 2004

Keywords

• easy
• nonn
• tabf

Comments

The number of members in the n-th row appears to be equal to 2 + ( (n) * ((1 + sqrt(5))/2) ), or the n-th member of the lower Wythoff sequence (A000201) plus two. For the four rows show above, these values are 3, 5, 6, 8.

The first member of each row n is the number of primes in the set {2^n...2^(n + 1) - 1} (sequence A036378). The last member of each row follows sequence A092097, which is also equivalent to taking the difference of successive members of A052130 (the number of products of half-odd primes less than 2^n).

Examples

			Let x = 1. In set {2^2..2^(3) - 1}, or {4, 5, 6, 7}, S{n - 1} = S{2 - 1} = S{1} = subset of all numbers with one factor (the primes). The size of this subset is 2, or {5, 7}. For the set {2^3...2^(4) - 1}, the size of subset S{3 - 1} is 4. For {2^4..2^(5) - 1}, the size of subset S{4 - 1} is 5. For all subsequent sets, the size of subset S{n - 1} will be 5.
The triangle begins:
  2,4,5
  2,4,6,7,8
  5,12,17,20,21,22
  7,20,30,37,41,44,46,47
  ...
		

Crossrefs

Cf. A000201, A036378, A092097, A052130.