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A216090 Numbers n such that k^(n-1) == k (mod n) for every k = 1, 2, ..., n-1. 2
1, 2, 6, 10, 14, 22, 26, 30, 34, 38, 46, 58, 62, 74, 82, 86, 94, 106, 118, 122, 134, 142, 146, 158, 166, 178, 182, 194, 202, 206, 214, 218, 226, 254, 262, 274, 278, 298, 302, 314, 326, 334, 346, 358, 362, 382, 386, 394, 398, 422, 446, 454, 458, 466, 478, 482 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,2

COMMENTS

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, …

Squarefree numbers n such that p-1 divides n-2 for every prime divisor p of n. - Paolo P. Lava, Feb 12 2014

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

LINKS

Paolo P. Lava, Table of n, a(n) for n = 1..1000

EXAMPLE

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

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

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

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

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

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

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

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

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

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

MAPLE

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:

[Alternative Maple program]

with(numtheory); P:=proc(i, j, k) local d, n, ok, p;

for n from 1 to i do p:=ifactors(n)[2]; ok:=1; for d from 1 to nops(p) do

if p[d][2]>1 or p[d][1]=j then ok:=0; break; fi;

if not type((n+k)/(p[d][1]+j), integer) then ok:=0; break; fi; od;

if ok=1 then print(n); fi; od; end: P(10^9, -1, -2); # Paolo P. Lava, Feb 12 2014

MATHEMATICA

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

PROG

(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

(Python)

from sympy.ntheory.factor_ import core

from sympy import primefactors

def ok(n):

if n<3: return True

if core(n) == n:

for p in primefactors(n):

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

return True

return False

print([n for n in range(1, 501) if ok(n)]) # Indranil Ghosh, Apr 23 2017

CROSSREFS

Subsequence of A192109.

Terms > 2 form a subsequence of A050990.

Cf. A197929, A197930, A197943.

Sequence in context: A039956 A197930 A192109 * A342641 A118369 A226829

Adjacent sequences: A216087 A216088 A216089 * A216091 A216092 A216093

KEYWORD

nonn

AUTHOR

Michel Lagneau, Sep 01 2012

STATUS

approved

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Last modified December 5 06:35 EST 2022. Contains 358582 sequences. (Running on oeis4.)