A322120 -id:A322120 - cp's OEIS frontend

A322121 Composite numbers m such that b^(m-1) == 1 (mod (b^2-1)*m) has a solution b.

25, 49, 65, 85, 91, 121, 125, 133, 145, 169, 185, 205, 217, 221, 247, 259, 265, 289, 301, 305, 325, 341, 343, 361, 365, 377, 403, 425, 427, 445, 451, 469, 481, 485, 493, 505, 511, 529, 533, 545, 553, 559, 565, 589, 625, 629, 637, 671, 679, 685, 689, 697, 703

Offset: 1

Thomas Ordowski, Nov 27 2018

nonn

The smallest solutions b are 7, 18, 8, 13, 3, 3, 57, 11, ...
These numbers m are odd and indivisible by 3.
They contain all prime powers p^k for p > 3 and k > 1.
It seems that, for a fixed integer k > 0, these are composite numbers m such that c^(m-1) == 1 (mod (c^2-1)m^k) for some base c.
Conjecture: If m is a composite number such that b^(m-1) == 1 (mod (b^2-1)m) for some base b, then m is a strong pseudoprime to some base a in the range 2 <= a <= m-2. Thus, these numbers m are probably a proper subset of A181782.

Cf. A181782, A322120.

aQ[n_] := CompositeQ[n] && LengthWhile[Range[2,n], !Divisible[#^(n-1)-1, (#^2-1) n] &] != n-1; Select[Range[1000],aQ] (* Amiram Eldar, Nov 27 2018 *)

More terms from Amiram Eldar, Nov 27 2018

cp's OEIS Frontend

A322121 Composite numbers m such that b^(m-1) == 1 (mod (b^2-1)*m) has a solution b.

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