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

Author

Karsten Meyer, Nov 02 2005

Keywords

• nonn

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

Examples

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

Links

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

Crossrefs

Cf. A002997, A039772, A209211, A227180.

Programs

Mathematica

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

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

Extensions

Edited by Don Reble, May 16 2006