A200046 -id:A200046 - cp's OEIS frontend

A200219 Number of solutions of the equation x^n + (x+1)^n = (x+2)^n (mod n) for x = 0..n-1.

1, 1, 1, 2, 1, 2, 1, 4, 3, 2, 1, 4, 1, 2, 0, 8, 1, 6, 1, 4, 1, 2, 1, 8, 0, 2, 9, 2, 1, 4, 1, 16, 0, 2, 0, 12, 1, 2, 0, 8, 1, 4, 1, 2, 3, 2, 1, 16, 7, 10, 2, 2, 1, 18, 0, 8, 0, 2, 1, 8, 1, 2, 3, 32, 2, 4, 1, 4, 0, 2, 1, 24, 1, 2, 0, 4, 6, 4, 1, 16, 27, 2, 1, 8

Offset: 1

Michel Lagneau, Nov 14 2011

nonn

a(n) = 0 for n = 15, 25, 33, 35, 39, 55, 57,… (see A200046).

a(n) = 1 if n prime.

			a(6) = 2 because:
for x = 3,  3^6 + 4^6 == 1(mod 6) and 5^6 == 1(mod 6).
for x = 5,  5^6 + 6^6 == 1 (mod 6) and (7)^6 == 1 (mod 6).

Michel Lagneau, Table of n, a(n) for n = 1..1000

Cf. A195637, A200046.

for n from 1 to 100 do:ii:=0:for x from 0 to n-1 do:if x^n+(x+1)^n -(x+2)^n mod n=0 then ii:=ii+1:else fi:od: printf(`%d, `,ii):od:

Array[Function[n,Count[Array[Mod[#^n+(#+1)^n-(#+2)^n,n]&,n,0],0]],84]

A200323 For each composite m = A002808(n), a(n) is the smallest number k for which the equation x^m + (x+k)^m = (x+k+1)^m (mod m) has no solution, where x = 0..m-1.

Original entry on oeis.org

2, 3, 2, 3, 3, 2, 2, 1, 2, 3, 2, 7, 2, 2, 1, 2, 3, 2, 3, 2, 1, 2, 1, 2, 3, 1, 2, 3, 2, 3, 4, 2, 2, 3, 2, 2, 3, 1, 2, 1, 3, 2, 2, 3, 2, 4, 3, 2, 1, 3, 2, 2, 1, 2, 2, 2, 2, 3, 2, 2, 3, 3, 2, 2, 3, 3, 2, 3, 4, 1, 2, 2, 1, 2, 3, 2, 7, 3, 2, 3, 2, 2, 3, 1, 2, 1, 2

Offset: 1

Michel Lagneau, Nov 16 2011

nonn

			a(12) = 7 because A002808(12) = 21 and the equation x^21 + (x+7)^21 = (x+8)^21 (mod 21)has no solution.

Cf. A002808, A200046.

for n from 1 to 120 do: i:=0:for k from 1 to 500 while(i=0) do :ii:=0:for x from 0 to n-1 do:if x^n+(x+k)^n -(x+k+1)^n mod n =0 then ii:=ii+1:else fi:od: if ii=0 then i:=1:printf(`%d, `,k):else fi:od:od:

cp's OEIS Frontend

A200219 Number of solutions of the equation x^n + (x+1)^n = (x+2)^n (mod n) for x = 0..n-1.

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