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

1,1

Comments

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.

Links

Table of n, a(n) for n=1..53.

Mathematica

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 *)

Crossrefs

Cf. A181782, A322120.

Sequence in context: A112771 A278931 A181782 * A020146 A020244 A020196

Adjacent sequences: A322118 A322119 A322120 * A322122 A322123 A322124

Keyword

nonn

Author

Thomas Ordowski, Nov 27 2018

Extensions

More terms from Amiram Eldar, Nov 27 2018

Status

approved