login
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.
0

%I #11 Nov 01 2021 13:40:07

%S 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,

%T 91,96,99,101,102,103,23,75,131,173,199,215,224,229,232,233,234,43,

%U 147,257,344,403,439,461,473,482,487,490,492,493

%N 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.

%C 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.

%C 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).

%e 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.

%e The triangle begins:

%e 2,4,5

%e 2,4,6,7,8

%e 5,12,17,20,21,22

%e 7,20,30,37,41,44,46,47

%e ...

%Y Cf. A000201, A036378, A092097, A052130.

%K easy,nonn,tabf

%O 1,1

%A _Andrew S. Plewe_, Jun 29 2004