A202034 -id:A202034 - cp's OEIS frontend

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

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

A340806 a(n) = Sum_{k=1..n-1} (k^n mod n).

Original entry on oeis.org

0, 1, 3, 2, 10, 13, 21, 4, 27, 45, 55, 38, 78, 77, 105, 8, 136, 93, 171, 146, 210, 209, 253, 172, 250, 325, 243, 294, 406, 365, 465, 16, 528, 561, 595, 402, 666, 665, 741, 372, 820, 673, 903, 726, 945, 897, 1081, 536, 1029, 1125, 1275, 1170, 1378, 765, 1485

Offset: 1

Sebastian Karlsson, Jan 22 2021

nonn

Sebastian Karlsson, Table of n, a(n) for n = 1..10000

Cf. A010848, A048153, A048155, A094264, A121706, A195637, A195812, A202034.

a:= n-> add(k&^n mod n, k=1..n-1):
seq(a(n), n=1..55);  # Alois P. Heinz, Feb 13 2021

a(n) = sum(k=1, n-1, lift(Mod(k, n)^n)); \\ Michel Marcus, Jan 22 2021

def a(n):
    return sum([pow(k,n,n) for k in range(1, n)])
for n in range(1, 56):
    print(a(n), end=', ')

a(n) = n*A010848(n)/2, if n is odd.
a(n) = n*(n-1)/2, if n is both odd and squarefree.
a(p^e) = (1/2)*(p-1)*p^(2*e-1), if p is an odd prime.
a(2^e) = 2^(e-1).

cp's OEIS Frontend

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

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

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A340806 a(n) = Sum_{k=1..n-1} (k^n mod n).

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