The latest mersenne exponent is added below.
position p in p # of bit p in hex p mod 4
in list base 10 in base 2 1's places
1 2 10 1 2 2 2
2 3 11 2 2 3 3
3 5 101 2 3 5 1
4 7 111 3 3 7 3
5 13 1101 3 4 D 3
6 17 10001 2 5 11 1
7 19 10011 3 5 13 3
8 31 11111 5 5 1F 3
9 61 111101 5 6 3D 1
10 89 1011001 4 7 59 1
11 107 1101011 5 7 6B 3
12 127 1111111 7 7 7F 3
13 521 1000001001 3 10 209 1
14 607 1001011111 7 10 25F 3
15 1279 10011111111 9 11 4FF 3
16 2203 100010011011 6 12 89B 3
17 2281 100011101001 6 12 8E9 1
18 3217 110010010001 5 12 C91 1
19 4253 1000010011101 6 13 109D 1
20 4423 1000101000111 6 13 1147 3
21 9689 10010111011001 8 14 25D9 1
22 9941 10011011010101 8 14 26D5 1
23 11213 10101111001101 9 14 2BCD 1
24 19937 100110111100001 8 15 4DE1 1
25 21701 101010011000101 7 15 54C5 1
26 23209 101101010101001 8 15 5AA9 1
27 44497 1010110111010001 9 16 ADD1 1
28 86243 10101000011100011 8 17 150E3 3
29 110503 11010111110100111 12 17 1AFA7 3
30 132049 100000001111010001 7 18 203D1 1
31 216091 110100110000011011 9 18 34C1B 3
32 756839 10111000110001100111 11 20 B8C67 3
33 859433 11010001110100101001 10 20 D1D29 1
34 1257787 100110011000100111011 11 21 13313B 3
35 1398269 101010101010111111101 14 21 1555FD 1
36? 2976221 1011010110100111011101 14 22 2D69DD 1
37? 3021377 1011100001101001000001 9 22 2E1A41 1
38? 6972593 11010100110010010110001 11 23 6A64B1 1
total 263 471 # 1's=21
# 3's=16
We'd expect the leading and trailing bits to be 1's except for
the sole even, accounting for 75 1 bits and 76 places. The
remaining "interior" bits are 1's 188 times out of 395, or 47.6%.
Up through 3021377 it was 47.9%.
With 2976221 it was 48.6% of the time. Without M2976221 it was 47.9%.
The incidence of 1's in the second place from the right (excluding
p=2) is 16/(16+21)=43.2%;
the incidence in the remaining interior bits is 172/358=48.0%
Largest run of 1's: 8 (p=1279)
Largest run of 0's: 7 (p=132049)
Highest percentage of 1's: 100% (p=3,7,31,127)
Lowest percentage of 1's: 30% (p=521; just one more bit than the minimum)
Ken
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