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

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

Adjacent sequences: A111302 A111303 A111304 * A111306 A111307 A111308

Keyword

nonn

Author

Karsten Meyer, Nov 02 2005

Extensions

Edited by Don Reble, May 16 2006

Status

approved