numerators

[Artur]

Numerators of n-th fraction of partial sum van der Waerden-Ulam binary  
measure of the primes

Conjecture:
if set of that same odd number occured time number is divisable by 2 with  
exception first 3 111

Any comments ?

ARTUR

Table[Numerator[PrimePi[k]/2^k], {k, 2, 1000}]
1, 1, 1, 3, 3, 1, 1, 1, 1, 5, 5, 3, 3, 3, 3, 7, 7, 1, 1, 1, 1, 9, 9, 9, 9,  
9, \
9, 5, 5, 11, 11, 11, 11, 11, 11, 3, 3, 3, 3, 13, 13, 7, 7, 7, 7, 15, 15,  
15, \
15, 15, 15, 1, 1, 1, 1, 1, 1, 17, 17, 9, 9, 9, 9, 9, 9, 19, 19, 19, 19, 5,  
5, \
21, 21, 21, 21, 21, 21, 11, 11, 11, 11, 23, 23, 23, 23, 23, 23, 3, 3, 3,  
3, \
3, 3, 3, 3, 25, 25, 25, 25, 13, 13, 27, 27, 27, 27, 7, 7, 29, 29, 29, 29,  
15, \
15, 15, 15, 15, 15, 15, 15, 15, 15, 15, 15, 15, 15, 31, 31, 31, 31, 1, 1,  
1, \
1, 1, 1, 33, 33, 17, 17, 17, 17, 17, 17, 17, 17, 17, 17, 35, 35, 9, 9, 9,  
9, \
9, 9, 37, 37, 37, 37, 37, 37, 19, 19, 19, 19, 39, 39, 39, 39, 39, 39, 5,  
5, \
5, 5, 5, 5, 41, 41, 21, 21, 21, 21, 21, 21, 21, 21, 21, 21, 43, 43, 11,  
11, \
11, 11, 45, 45, 23, 23, 23, 23, 23, 23, 23, 23, 23, 23, 23, 23, 47, 47,  
47, \
47, 47, 47, 47, 47, 47, 47, 47, 47, 3, 3, 3, 3, 49, 49, 25, 25, 25, 25,  
51, \
51, 51, 51, 51, 51, 13, 13, 53, 53, 53, 53, 53, 53, 53, 53, 53, 53, 27,  
27, \
27, 27, 27, 27, 55, 55, 55, 55, 55, 55, 7, 7, 7, 7, 7, 7, 57, 57, 29, 29,  
29, \
29, 29, 29, 59, 59, 59, 59, 15, 15, 61, 61, 61, 61, 61, 61, 61, 61, 61,  
61, \
31, 31, 31, 31, 31, 31, 31, 31, 31, 31, 31, 31, 31, 31, 63, 63, 63, 63, 1,  
1, \
65, 65, 65, 65, 33, 33, 33, 33, 33, 33, 33, 33, 33, 33, 33, 33, 33, 33,  
67, \
67, 67, 67, 67, 67, 17, 17, 17, 17, 17, 17, 17, 17, 17, 17, 69, 69, 35,  
35, \
35, 35, 71, 71, 71, 71, 71, 71, 9, 9, 9, 9, 9, 9, 9, 9, 73, 73, 73, 73,  
73, \
73, 37, 37, 37, 37, 37, 37, 75, 75, 75, 75, 19, 19, 19, 19, 19, 19, 77,  
77, \
77, 77, 77, 77, 77, 77, 39, 39, 39, 39, 79, 79, 79, 79, 79, 79, 79, 79, 5,  
5, \
5, 5, 5, 5, 5, 5, 5, 5, 81, 81, 41, 41, 41, 41, 41, 41, 41, 41, 41, 41,  
83, \
83, 21, 21, 21, 21, 21, 21, 85, 85, 85, 85, 43, 43, 43, 43, 43, 43, 87,  
87, \
87, 87, 87, 87, 87, 87, 11, 11, 11, 11, 89, 89, 45, 45, 45, 45, 91, 91,  
91, \
91, 91, 91, 91, 91, 91, 91, 91, 91, 23, 23, 23, 23, 23, 23, 23, 23, 93,  
93, \
93, 93, 47, 47, 47, 47, 47, 47, 47, 47, 95, 95, 95, 95, 3, 3, 3, 3, 3, 3,  
97, \
97, 97, 97, 97, 97, 97, 97, 97, 97, 97, 97, 49, 49, 99, 99, 99, 99, 99,  
99, \
99, 99, 99, 99, 99, 99, 99, 99, 99, 99, 99, 99, 25, 25, 25, 25, 25, 25,  
101, \
101, 101, 101, 101, 101, 101, 101, 101, 101, 51, 51, 51, 51, 51, 51, 103, \
103, 103, 103, 103, 103, 13, 13, 105, 105, 105, 105, 105, 105, 53, 53, 53,  
\
53, 53, 53, 53, 53, 53, 53, 107, 107, 107, 107, 107, 107, 27, 27, 27, 27,  
27, \
27, 109, 109, 55, 55, 55, 55, 55, 55, 111, 111, 111, 111, 111, 111, 7, 7,  
7, \
7, 113, 113, 57, 57, 57, 57, 57, 57, 57, 57, 57, 57, 57, 57, 115, 115,  
115, \
115, 115, 115, 115, 115, 115, 115, 29, 29, 117, 117, 117, 117, 59, 59, 59,  
\
59, 59, 59, 119, 119, 119, 119, 119, 119, 15, 15, 121, 121, 121, 121, 121,  
\
121, 121, 121, 121, 121, 121, 121, 61, 61, 61, 61, 123, 123, 123, 123,  
123, \
123, 31, 31, 31, 31, 31, 31, 31, 31, 125, 125, 125, 125, 125, 125, 125,  
125, \
125, 125, 63, 63, 63, 63, 63, 63, 63, 63, 127, 127, 127, 127, 127, 127,  
127, \
127, 127, 127, 1, 1, 1, 1, 1, 1, 1, 1, 129, 129, 129, 129, 129, 129, 65,  
65, \
65, 65, 65, 65, 131, 131, 131, 131, 33, 33, 33, 33, 33, 33, 33, 33, 133,  
133, \
133, 133, 133, 133, 67, 67, 67, 67, 135, 135, 135, 135, 135, 135, 135,  
135, \
17, 17, 17, 17, 137, 137, 137, 137, 137, 137, 137, 137, 137, 137, 137,  
137, \
137, 137