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A322121 Composite numbers m such that b^(m-1) == 1 (mod (b^2-1)*m) has a solution b. 1
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 (list; graph; refs; listen; history; text; internal format)
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

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Last modified May 16 14:39 EDT 2021. Contains 343949 sequences. (Running on oeis4.)