

A322121


Composite numbers m such that b^(m1) == 1 (mod (b^21)*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
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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^(m1) == 1 (mod (c^21)m^k) for some base c.
Conjecture: If m is a composite number such that b^(m1) == 1 (mod (b^21)m) for some base b, then m is a strong pseudoprime to some base a in the range 2 <= a <= m2. Thus, these numbers m are probably a proper subset of A181782.


LINKS



MATHEMATICA

aQ[n_] := CompositeQ[n] && LengthWhile[Range[2, n], !Divisible[#^(n1)1, (#^21) n] &] != n1; Select[Range[1000], aQ] (* Amiram Eldar, Nov 27 2018 *)


CROSSREFS



KEYWORD

nonn


AUTHOR



EXTENSIONS



STATUS

approved



