

A239452


Smallest integer m > 1 such that m^n == m (mod n).


4



2, 2, 2, 4, 2, 3, 2, 8, 8, 5, 2, 4, 2, 7, 4, 16, 2, 9, 2, 5, 6, 11, 2, 9, 7, 13, 26, 4, 2, 6, 2, 32, 10, 17, 6, 9, 2, 19, 12, 16, 2, 7, 2, 12, 8, 23, 2, 16, 18, 25, 16, 9, 2, 27, 10, 8, 18, 29, 2, 16, 2, 31, 8, 64, 5, 3, 2, 17, 22, 11, 2, 9, 2, 37, 24, 20, 21
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OFFSET

1,1


COMMENTS

Composite n are Fermat weak pseudoprimes to base a(n).
If n > 2 is prime then a(n) = 2. The converse is false : a(341) = 2 and 341 isn't prime.
a(n) <= A105222(n). a(n) = A105222(n) if and only if a(n) is coprime to n.
For n > 1, a(n) <= n and if a(n) = n, then A105222(n) = n+1.
It seems that a(n) = n if and only if n = 2^k with k > 0, a(n) = n1 if and only if n = 3^k with k > 0, a(2n) = n if and only if n = p^k where p is an odd prime and k > 0.  Thomas Ordowski, Oct 19 2017


LINKS

Giovanni Resta, Table of n, a(n) for n = 1..10000
GĂ©rard P. Michon, Weak pseudoprimes to base a


EXAMPLE

We have 2^4 != 2, 3^4 != 3, but 4^4 == 4 (mod 4), so a(4) = 4.


MAPLE

L:=NULL:for n to 100 do for a from 2 while a^n  a mod n !=0 do od; L:=L, a od: L;


MATHEMATICA

a[n_] := Block[{m = 2}, While[PowerMod[m, n, n] != Mod[m, n], m++]; m]; Array[a, 100] (* Giovanni Resta, Mar 19 2014 *)


PROG

(Haskell)
import Math.NumberTheory.Moduli (powerMod)
a239452 n = head [m  m < [2..], powerMod m n n == mod m n]
 Reinhard Zumkeller, Mar 19 2014
(Python)
L=[];
for n in range(1, 101):
...a=2
...while (a**n  a) % n != 0:
......a+=1
...L=L+[a]
L
(PARI) a(n)=my(m=2); while(Mod(m, n)^n!=m, m++); m \\ Charles R Greathouse IV, Mar 21 2014


CROSSREFS

Cf. A105222.
Sequence in context: A029640 A029658 A332889 * A069930 A086327 A114896
Adjacent sequences: A239449 A239450 A239451 * A239453 A239454 A239455


KEYWORD

nonn


AUTHOR

Robert FERREOL, Mar 19 2014


EXTENSIONS

a(20)a(77) from Giovanni Resta, Mar 19 2014


STATUS

approved



