A216090 - OEIS

%I #38 Mar 16 2024 11:19:50

%S 1,2,6,10,14,22,26,30,34,38,46,58,62,74,82,86,94,106,118,122,134,142,

%T 146,158,166,178,182,194,202,206,214,218,226,254,262,274,278,298,302,

%U 314,326,334,346,358,362,382,386,394,398,422,446,454,458,466,478,482

%N Numbers n such that k^(n-1) == k (mod n) for every k = 1, 2, ..., n-1.

%C Subsequence of, but different from A197930, for example A197930(11) = 42 with 42 distinct residues, but the set R of the residues k^41 mod 42 is R = {1, 32, 33, 16, 17, 6, …, 9, 10, 41} for k = 1, 2, …, 41 instead  R = {1, 2, 3, …, 40, 41}. Terms of A197930 that are not in this sequence: 42, 78, 110, 114, 138, 170, …

%C Squarefree numbers n such that A002322(n) divides n-2. Contains all doubled odd primes and all doubled Carmichael numbers. - _Thomas Ordowski_, Apr 23 2017

%H Paolo P. Lava, <a href="/A216090/b216090.txt">Table of n, a(n) for n = 1..1000</a>

%e a(4) = 10 because x^9  == 1, 2, ..., 9  (mod 10) with 9 distinct residues such that:

%e 1^9 = 1 == 1 (mod 10);

%e 2^9 = 512 == 2 (mod 10);

%e 3^9 = 19683 == 3 (mod 10);

%e 4^9 = 262144 == 4 (mod 10);

%e 5^9 = 1953125 == 5 (mod 10);

%e 6^9 = 10077696 == 6 (mod 10);

%e 7^9 = 40353607 == 7 (mod 10);

%e 8^9 = 134217728 == 8 (mod 10);

%e 9^9 = 387420489 == 9 (mod 10).

%p with(numtheory):for n from 1  to 500 do:j:=0:for i from 1 to n do: if irem(i^(n-1),n)=i then j:=j+1:else fi:od:if j=n-1 then printf(`%d, `, n):else fi:od:

%t f[n_] := And @@ Table[PowerMod[k, n - 1, n] == k, {k, n - 1}]; Select[Range[500], f] (* _T. D. Noe_, Sep 03 2012 *)

%o (PARI) isok(n) = {for (k=1, n-1, if (Mod(k, n)^(n-1) != Mod(k, n), return (0));); return (1);} \\ _Michel Marcus_, Apr 23 2017

%o (Python)

%o from sympy.ntheory.factor_ import core

%o from sympy import primefactors

%o def ok(n):

%o     if n<3: return True

%o     if core(n) == n:

%o         for p in primefactors(n):

%o             if (n - 2)%(p - 1): return False

%o         return True

%o     return False

%o print([n for n in range(1, 501) if ok(n)]) # _Indranil Ghosh_, Apr 23 2017

%Y Subsequence of A192109.

%Y Terms > 2 form a subsequence of A050990.

%Y Cf. A197929, A197930, A197943.

%K nonn

%O 1,2

%A _Michel Lagneau_, Sep 01 2012