OFFSET
1,1
COMMENTS
A majority of numbers k satisfies the equation n^k mod (2*k+1) = k^n mod (2*n+1) = r = 1.
The values of n such that r <> 1 are given by n = 17, 38, 42, 47, 57, 59, ..., including the values with r = 0 given by n = 62, 84, 171, ...
LINKS
Paolo Xausa, Table of n, a(n) for n = 1..10000
EXAMPLE
a(5) = 9 because 5^9 mod 19 = 9^5 mod 11 = 1;
a(17) = 5 because 17^5 mod 11 = 5^17 mod 35 = 10;
a(62) = 15 because 62^15 mod 31 = 15^62 mod 125 = 0.
MAPLE
with(numtheory): for n from 1 to 100 do:ii:=0:for k from 1 to 10000 while(ii=0) do:if n<>k and irem(n^k, 2*k+1) = irem(k^n, 2*n+1) then ii:=1:printf(`%d, `, k):else fi:od:od:
MATHEMATICA
A210693[n_] := Module[{k = 0}, While[++k == n || PowerMod[n, k, 2*k + 1] != PowerMod[k, n, #]] & [2*n + 1]; k];
Array[A210693, 100] (* Paolo Xausa, Dec 20 2025 *)
CROSSREFS
KEYWORD
nonn
AUTHOR
Michel Lagneau, Mar 30 2012
STATUS
approved
