A197929 -id:A197929 - cp's OEIS frontend

A197930 Numbers n such that the number of distinct residues in x^(n-1) (mod n), x=0..n-1, equals n.

1, 2, 6, 10, 14, 22, 26, 30, 34, 38, 42, 46, 58, 62, 74, 78, 82, 86, 94, 102, 106, 110, 114, 118, 122, 134, 138, 142, 146, 158, 166, 170, 174, 178, 182, 194, 202, 206, 210, 214, 218, 222, 226, 230, 254, 258, 262, 266, 274, 278, 282, 290, 298, 302, 314, 318

Offset: 1

Michel Lagneau, Oct 19 2011

nonn

a(n) = n if n = 2p, p prime > 2, or n = 2q with q nonprime such that q = 1, 15, 21, 39, 51, 55, 57, 69, 85, 87, 91,…

			a(8) = 30 because x^29  == 0,1,2, …,28,29  (mod 30) with 30 distinct residues.

Harvey P. Dale, Table of n, a(n) for n = 1..1000

Cf. A197929, A195637.

lst={}; Table[If[Length[Union[PowerMod[Range[0,n-1],n-1,n]]]==n, AppendTo[lst,n]], {n,320}]; lst
Select[Range[400],Length[Union[PowerMod[Range[0,#-1],#-1,#]]]==#&] (* Harvey P. Dale, Nov 06 2016 *)

n such that A197929(n) = n.

A198020 Number of distinct residues of x^n (mod 2n+1), x=0..2n.

Original entry on oeis.org

1, 3, 3, 3, 4, 3, 3, 15, 3, 3, 8, 3, 6, 19, 3, 3, 12, 35, 3, 39, 3, 3, 12, 3, 8, 51, 3, 55, 20, 3, 3, 49, 8, 3, 24, 3, 3, 63, 24, 3, 28, 3, 27, 87, 3, 15, 32, 95, 3, 77, 3, 3, 16, 3, 3, 111, 3, 115, 28, 119, 12, 123, 51, 3, 44, 3, 8, 95, 3, 3, 48, 143, 16, 129

Offset: 0

Michel Lagneau, Oct 20 2011

nonn

a(n) = 3 if 2n+1 prime because the corresponding residues are 0, 1 and 2n (mod 2n+1).

			a(7) = 15 because x^7  == 0, 1, …,14  (mod 15) => 15 distinct residues.

Amiram Eldar, Table of n, a(n) for n = 0..10000

Cf. A195637, A197929.

Table[Length[Union[PowerMod[Range[0, 2*n], n, 2*n+1]]], {n,0, 100}]

A197943 Greatest residue of x^(n-1) (mod n), x=0..n-1.

Original entry on oeis.org

0, 1, 1, 3, 1, 5, 1, 7, 7, 9, 1, 11, 1, 13, 10, 15, 1, 17, 1, 19, 18, 21, 1, 23, 21, 25, 25, 27, 1, 29, 1, 31, 31, 33, 30, 35, 1, 37, 36, 39, 1, 41, 1, 43, 40, 45, 1, 47, 43, 49, 49, 51, 1, 53, 49, 55, 55, 57, 1, 59, 1, 61, 58, 63, 61, 65, 1, 67, 64, 69, 1, 71, 1, 73, 69, 75, 71, 77, 1, 79, 79, 81, 1, 83, 81, 85, 82, 87, 1, 89, 78

Offset: 1

Michel Lagneau, Oct 19 2011

a(n) = 1 if n prime and a(n) = n-1 if n even.

			a(8) = 7 because x^7 == 0, 1, 3, 5, 7  (mod 8) => 7 is the greatest residue.

Antti Karttunen, Table of n, a(n) for n = 1..11111

Cf. A196082, A197929, A197930.

Table[Max[PowerMod[Range[0,n-1],n-1,n]], {n,100}]

A197943(n) = { my(m=0); for(x=0,n-1, m = max(m,lift(Mod(x^(n-1),n)))); (m); }; \\ Antti Karttunen, Sep 10 2018

More terms added, incorrect PARI-program removed by Antti Karttunen, Sep 10 2018

A216090 Numbers n such that k^(n-1) == k (mod n) for every k = 1, 2, ..., n-1.

Original entry on oeis.org

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

Offset: 1

Michel Lagneau, Sep 01 2012

nonn

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 A002322(n) divides n-2. Contains all doubled odd primes and all doubled Carmichael numbers. - Thomas Ordowski, Apr 23 2017

			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).

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

Subsequence of A192109.
Terms > 2 form a subsequence of A050990.
Cf. A197929, A197930, A197943.

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:

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

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

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

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A197930 Numbers n such that the number of distinct residues in x^(n-1) (mod n), x=0..n-1, equals n.

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A198020 Number of distinct residues of x^n (mod 2n+1), x=0..2n.

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A197943 Greatest residue of x^(n-1) (mod n), x=0..n-1.

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A216090 Numbers n such that k^(n-1) == k (mod n) for every k = 1, 2, ..., n-1.

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