A196082 -id:A196082 - cp's OEIS frontend

A196499 Numbers k such that the greatest residue of the congruence x^k (mod k) equals k-1 for x in [0..k-1].

1, 2, 3, 5, 7, 9, 10, 11, 13, 15, 17, 19, 21, 23, 25, 26, 27, 29, 31, 33, 34, 35, 37, 39, 41, 43, 45, 47, 49, 50, 51, 53, 55, 57, 58, 59, 61, 63, 65, 67, 69, 71, 73, 74, 75, 77, 79, 81, 82, 83, 85, 87, 89, 91, 93, 95, 97, 99, 101, 103, 105, 106, 107, 109, 111

Offset: 1

Michel Lagneau, Oct 13 2011

nonn

All prime numbers are in the sequence.

			50 is in the sequence because the residues  of the congruence x^50 (mod 50) are {0, 1, 24, 25, 26, 49} and the greatest value is 50 - 1 = 49.

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A196082.

lst := {}; Do[If[Max[Union[PowerMod[Range[0, n - 1], n, n]]] == n - 1, AppendTo[lst, n]], {n, 120}]; lst

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

A198033 Greatest residue of x^n (mod 2n+1), x=0..2n.

Original entry on oeis.org

0, 2, 4, 6, 7, 10, 12, 14, 16, 18, 18, 22, 21, 26, 28, 30, 31, 34, 36, 38, 40, 42, 40, 46, 43, 50, 52, 54, 55, 58, 60, 62, 61, 66, 64, 70, 72, 74, 71, 78, 79, 82, 84, 86, 88, 90, 90, 94, 96, 98, 100, 102, 100, 106, 108, 110, 112, 114, 108, 118, 111, 122, 124

Offset: 0

Michel Lagneau, Oct 20 2011

nonn

If 2n+1 is prime, a(n) = 2n. But there exists nonprime numbers of the form 2n+1 such that a(n) = 2n, for example n = 0, 7, 13, 17, 19, 25, 27, 31, 37, 42, …

			a(10) = 18 because x^10 == 0, 1, 4, 7, 9, 15, 16, 18 (mod 21) => 18 is the greatest residue.

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

Cf. A196082, A198020, A198032.

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

A202035 Greatest prime residue of x^n (mod n) for x=0..n-1, or 0 if no such prime exists.

Original entry on oeis.org

0, 0, 2, 0, 3, 3, 5, 0, 0, 5, 7, 0, 11, 11, 13, 0, 13, 0, 17, 5, 13, 11, 19, 0, 7, 23, 0, 0, 23, 19, 29, 0, 31, 19, 31, 0, 31, 23, 31, 0, 37, 7, 41, 37, 37, 41, 43, 0, 31, 0, 47, 29, 47, 0, 43, 0, 37, 53, 53, 0, 59, 59, 0, 0, 61, 37, 61, 17, 67, 29, 67, 0, 71

Offset: 1

Michel Lagneau, Dec 09 2011

nonn

			a(7) = 3  because  k^7 ==0, 1, 2, 3, 4, 5, 6 (mod 7) => 5 is the greatest prime.

Cf. A196082, A202034.

for n from 1 to 100 do: W:={}:for k from 0 to n-1 do:z:= irem(k^n,n): if type(z,prime)=true then W:=W union {z}:else fi:od: x:=nops(W): if x<>0 then printf(`%d, `,W[x]): else printf(`%d, `,0):fi: od:

A202036 Smallest prime residue of x^n (mod n) for x=0..n-1, or 0 if no such prime exists.

Original entry on oeis.org

0, 0, 2, 0, 2, 3, 2, 0, 0, 5, 2, 0, 2, 2, 2, 0, 2, 0, 2, 5, 7, 3, 2, 0, 7, 3, 0, 0, 2, 19, 2, 0, 2, 2, 2, 0, 2, 5, 5, 0, 2, 7, 2, 5, 17, 2, 2, 0, 19, 0, 2, 13, 2, 0, 11, 0, 7, 5, 2, 0, 2, 2, 0, 0, 2, 3, 2, 13, 2, 11, 2, 0, 2, 3, 7, 5, 2, 13, 2, 0, 0, 2, 2, 0

Offset: 1

Michel Lagneau, Dec 09 2011

nonn

			a(7) = 2 because k^7 == 0, 1, 2, 3, 4, 5, 6 (mod 7) => 2 is the smallest prime.

Antti Karttunen, Table of n, a(n) for n = 1..16384
Antti Karttunen, Data supplement: n, a(n) computed for n = 1..65537

Cf. A195812, A196082, A202034, A202035.

for n from 1 to 100 do: W:={}:for k from 0 to n-1 do:z:= irem(k^n,n): if type(z,prime)=true then W:=W union {z}:else fi:od: x:=nops(W): if x<>0 then printf(`%d, `,W[1]): else printf(`%d, `,0):fi: od:

Table[SelectFirst[Sort[PowerMod[Range[n-1],n,n]],PrimeQ],{n,90}]/.Missing["NotFound"]->0 (* Harvey P. Dale, May 01 2023 *)

A202036(n) = { my(z,y=n); for(x=1,n-1,z = lift(Mod(x,n)^n); if(isprime(z), y = min(z,y))); if(y==n,0,y); }; \\ - Antti Karttunen, May 19 2021

A197113 Nonprime numbers n such that the greatest residue of the congruence x^n (mod n) equals n-1 where x = 0..n-1.

Original entry on oeis.org

1, 9, 10, 15, 21, 25, 26, 27, 33, 34, 35, 39, 45, 49, 50, 51, 55, 57, 58, 63, 65, 69, 74, 75, 77, 81, 82, 85, 87, 91, 93, 95, 99, 105, 106, 111, 115, 117, 119, 121, 122, 123, 125, 129, 130, 133, 135, 141, 143, 145, 146, 147, 153, 155, 159, 161, 165, 169, 170

Offset: 1

Michel Lagneau, Oct 13 2011

nonn

Subset of A196499.

For all primes n, the greatest residue of the congruence x^n (mod n) where x = 0..n-1 equals n-1.

			50 is in the sequence because the residues of the congruence x^50 (mod 50) are { 0, 1, 24, 25, 26, 49} and the greatest value is 50 - 1 = 49.

Cf. A196499, A196082.

with(numtheory):T:=array(1..170): for n from 1 to 170 do:for k from 1 to n do:T[k]:=irem(k^n,n):od:W:=convert(T,set):x:=nops(W):if type(n,prime) = false and W[x]= n-1 then printf(`%d, `,n):else fi:od:

cp's OEIS Frontend

A196499 Numbers k such that the greatest residue of the congruence x^k (mod k) equals k-1 for x in [0..k-1].

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

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

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Extensions

A198033 Greatest residue of x^n (mod 2n+1), x=0..2n.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

A202035 Greatest prime residue of x^n (mod n) for x=0..n-1, or 0 if no such prime exists.

Views

Author

Keywords

Examples

Crossrefs

Programs

Maple

A202036 Smallest prime residue of x^n (mod n) for x=0..n-1, or 0 if no such prime exists.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Maple

Mathematica

PARI

A197113 Nonprime numbers n such that the greatest residue of the congruence x^n (mod n) equals n-1 where x = 0..n-1.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Maple