

A111305


Composite numbers n such that a^(n1) = 1 mod n only when a = 1 mod n.


2



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)^(n1) = 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 there because 3^9 = 3, 7^9 = 7, 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, n1, if(Mod(a, n)^(n1)==1, return(0))); !isprime(n) \\ Charles R Greathouse IV, Dec 22 2016


CROSSREFS

Cf. A002997, A039772, A209211, A227180.
Sequence in context: A163300 A193175 A093161 * 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



