check_note.py  (Python 3.13.5; nmax = 131072)
[14] Proposition 6.1: the certificates (a) and the certificates (b) for N <= 131072
    N =      4: |x_N| =      7, |sigma0(x_N)| =     13, ce(x_N) = 7/4           ce(sigma0(x_N)) = 5/3           (0.0 s)
    N =      6: |x_N| =     11, |sigma0(x_N)| =     21, ce(x_N) = 11/6          ce(sigma0(x_N)) = 7/4           (0.0 s)
    N =      8: |x_N| =     15, |sigma0(x_N)| =     29, ce(x_N) = 15/8          ce(sigma0(x_N)) = 11/6          (0.0 s)
    N =     12: |x_N| =     23, |sigma0(x_N)| =     45, ce(x_N) = 23/12         ce(sigma0(x_N)) = 15/8          (0.0 s)
    N =     16: |x_N| =     31, |sigma0(x_N)| =     61, ce(x_N) = 31/16         ce(sigma0(x_N)) = 23/12         (0.0 s)
    N =     24: |x_N| =     47, |sigma0(x_N)| =     93, ce(x_N) = 47/24         ce(sigma0(x_N)) = 31/16         (0.0 s)
    N =     32: |x_N| =     63, |sigma0(x_N)| =    125, ce(x_N) = 63/32         ce(sigma0(x_N)) = 47/24         (0.0 s)
    N =     48: |x_N| =     95, |sigma0(x_N)| =    189, ce(x_N) = 95/48         ce(sigma0(x_N)) = 63/32         (0.0 s)
    N =     64: |x_N| =    127, |sigma0(x_N)| =    253, ce(x_N) = 127/64        ce(sigma0(x_N)) = 95/48         (0.0 s)
    N =     96: |x_N| =    191, |sigma0(x_N)| =    381, ce(x_N) = 191/96        ce(sigma0(x_N)) = 127/64        (0.0 s)
    N =    128: |x_N| =    255, |sigma0(x_N)| =    509, ce(x_N) = 255/128       ce(sigma0(x_N)) = 191/96        (0.0 s)
    N =    192: |x_N| =    383, |sigma0(x_N)| =    765, ce(x_N) = 383/192       ce(sigma0(x_N)) = 255/128       (0.0 s)
    N =    256: |x_N| =    511, |sigma0(x_N)| =   1021, ce(x_N) = 511/256       ce(sigma0(x_N)) = 383/192       (0.0 s)
    N =    384: |x_N| =    767, |sigma0(x_N)| =   1533, ce(x_N) = 767/384       ce(sigma0(x_N)) = 511/256       (0.0 s)
    N =    512: |x_N| =   1023, |sigma0(x_N)| =   2045, ce(x_N) = 1023/512      ce(sigma0(x_N)) = 767/384       (0.0 s)
    N =    768: |x_N| =   1535, |sigma0(x_N)| =   3069, ce(x_N) = 1535/768      ce(sigma0(x_N)) = 1023/512      (0.0 s)
    N =   1024: |x_N| =   2047, |sigma0(x_N)| =   4093, ce(x_N) = 2047/1024     ce(sigma0(x_N)) = 1535/768      (0.0 s)
    N =   1536: |x_N| =   3071, |sigma0(x_N)| =   6141, ce(x_N) = 3071/1536     ce(sigma0(x_N)) = 2047/1024     (0.0 s)
    N =   2048: |x_N| =   4095, |sigma0(x_N)| =   8189, ce(x_N) = 4095/2048     ce(sigma0(x_N)) = 3071/1536     (0.1 s)
    N =   3072: |x_N| =   6143, |sigma0(x_N)| =  12285, ce(x_N) = 6143/3072     ce(sigma0(x_N)) = 4095/2048     (0.1 s)
    N =   4096: |x_N| =   8191, |sigma0(x_N)| =  16381, ce(x_N) = 8191/4096     ce(sigma0(x_N)) = 6143/3072     (0.2 s)
    N =   6144: |x_N| =  12287, |sigma0(x_N)| =  24573, ce(x_N) = 12287/6144    ce(sigma0(x_N)) = 8191/4096     (0.4 s)
    N =   8192: |x_N| =  16383, |sigma0(x_N)| =  32765, ce(x_N) = 16383/8192    ce(sigma0(x_N)) = 12287/6144    (0.8 s)
    N =  12288: |x_N| =  24575, |sigma0(x_N)| =  49149, ce(x_N) = 24575/12288   ce(sigma0(x_N)) = 16383/8192    (1.7 s)
    N =  16384: |x_N| =  32767, |sigma0(x_N)| =  65533, ce(x_N) = 32767/16384   ce(sigma0(x_N)) = 24575/12288   (3.0 s)
    N =  24576: |x_N| =  49151, |sigma0(x_N)| =  98301, ce(x_N) = 49151/24576   ce(sigma0(x_N)) = 32767/16384   (6.8 s)
    N =  32768: |x_N| =  65535, |sigma0(x_N)| = 131069, ce(x_N) = 65535/32768   ce(sigma0(x_N)) = 49151/24576   (12.0 s)
    N =  49152: |x_N| =  98303, |sigma0(x_N)| = 196605, ce(x_N) = 98303/49152   ce(sigma0(x_N)) = 65535/32768   (26.9 s)
    N =  65536: |x_N| = 131071, |sigma0(x_N)| = 262141, ce(x_N) = 131071/65536  ce(sigma0(x_N)) = 98303/49152   (47.7 s)
    N =  98304: |x_N| = 196607, |sigma0(x_N)| = 393213, ce(x_N) = 196607/98304  ce(sigma0(x_N)) = 131071/65536  (104.8 s)
    N = 131072: |x_N| = 262143, |sigma0(x_N)| = 524285, ce(x_N) = 262143/131072 ce(sigma0(x_N)) = 196607/98304  (193.2 s)
    the intervals (ce(sigma0(x_N)), ce(x_N)] chain; their union is (5/3, 262143/131072]
    (397.8 s)
TOTAL failures: 0   (running time 398 s)
