A111305 - OEIS

A111305

Composite numbers k such that a^(k-1) == 1 (mod k) only when a == 1 (mod k).

4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 30, 32, 34, 36, 38, 40, 42, 44, 46, 48, 50, 54, 56, 58, 60, 62, 64, 68, 72, 74, 78, 80, 82, 84, 86, 88, 90, 92, 94, 96, 98, 100, 102, 104, 106, 108, 110, 114, 116, 118, 120, 122, 126, 128, 132, 134, 136, 138, 140, 142, 144

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OFFSET

1,1

COMMENTS

These unCarmichael numbers fail the Fermat primality test as often as possible.

All such numbers are even: for odd n, (-1)^(n-1) = 1.

The even numbers not in this sequence are 2 and A039772.

If c is a Carmichael number, then 2c is in the sequence. Also, the sequence is A209211 without the first two terms. - Emmanuel Vantieghem, Jul 03 2013

LINKS

Table of n, a(n) for n=1..63.

Romeo Meštrović, Generalizations of Carmichael numbers I, arXiv:1305.1867v1 [math.NT], May 04 2013.

EXAMPLE

10 is a term because 3^9 == 3 (mod 10), 7^9 == 7 (mod 10), 9^9 == 9 (mod 10).

MATHEMATICA

Select[Range[4, 144], Count[Table[PowerMod[b, # - 1, #], {b, 1, # - 1}], 1] == 1 &] (* Geoffrey Critzer, Apr 11 2015 *)

PROG

(PARI) is(n)=for(a=2, n-1, if(Mod(a, n)^(n-1)==1, return(0))); !isprime(n) \\ Charles R Greathouse IV, Dec 22 2016

CROSSREFS

Cf. A002997, A039772, A209211, A227180.

Sequence in context: A307782 A325038 A360127 * A284665 A210939 A175246