[seqfan] (anti-Carmichael) Pseudoprimes n to base D_{n-1}

[Tomasz Ordowski]

Dear readers!

Let us define:
Odd numbers n > 1 such that a^{n-1} == 1 (mod n),
where a = D_{n-1} is the denominator of Bernoulli number B_{n-1}.
For odd n > 1, D_{n-1} = Product_{p prime, p-1|n-1} p = Product_{d|n-1, d+1
prime} (d+1).
35, 533, 869, 1247, 1271, 1513, 1807, 6943, 8473, 9211, 10849,
11453, 14315, 15523, 19729, 21463, 22399, 23377, 23651, 31291, 32437,
35711, 38491, 49489, 53941, 56575, 74563, 76201, 87001, 88183,
90157, 105821, 113815, 128627, 139057, 150851, 158237, 186979,
189409, 194737, 232831, 262417, 264215, 276841, 282829, 297541,
315169, 324137, 346837, 371659, 384041, 398801, 400387, 455891,
459709, 481793, 574841, 620621, 676549, 699679, 701893, 725869,
763129, 763813, 775321, 789427, 813451, 813793, 881809, 889733, 899083,
935953, 981317, ....
Problem: are there infinitely many such numbers?

Consider the following subset of these pseudoprimes.
Numbers m such that D_{m-1} is the smallest base b > 1 for which b^{m-1} ==
1 (mod m).
35, 14315,  22399, 35711, 455891, 881809, 1198159, 1917071, 2287987,
3310037, 4464941, 11029439, 12190061, 13325753, 17832803, 33012941,
33296147, 37814849, 44986423, 74437181, 76911149, 82873661, 91909571,
98859851, 108266171, 128008159, 128981243, 132391409, ...
These are numbers m such that A027642(m-1) = A105222(m).
Here are the corresponding values for the bases (from the above equality):
6, 6, 42, 66, 66, 46410, 3318, 66, 42, 30, 330, 6, 330, 61410, 6, 330,
1074, 510, 3318, 330, 7890, 330, 66, 12606, 66, 42, 6, 510, ...

I noticed that the number 35 has a rare property, namely:
for every prime p|(6^{35-1}-1)/35, we have 35-1|p-1.
Unfortunately, for other numbers found,
lpf((b^{n-1}-1)/n) < n.

Best regards,

Thomas Ordowski
______________________
All data from Amiram Eldar.
_____________________
See on the OEIS website:
"anti-Carmichael"
- numbers
A121707 - OEIS <https://oeis.org/A121707>
- pseudoprimes
A316940 - OEIS <https://oeis.org/A316940>