A216091 Numbers n such that k == k^(q-1) mod q for k = 1, 2, ..., q-1, where q = n^2+1.

1, 3, 5, 9, 11, 15, 19, 25, 29, 35, 39, 45, 47, 49, 51, 59, 61, 65, 69, 71, 79, 85, 95, 101, 121, 131, 139, 141, 145, 159, 165, 169, 171, 175, 181, 195, 199, 201, 205, 209, 219, 221, 231, 245, 261, 271, 275, 279, 289, 299, 309, 315, 321, 325, 329, 335, 345

Offset: 1

Author

Michel Lagneau, Sep 01 2012

Keywords

• nonn

Comments

It is interesting to note that this sequence is identical to A002731 except for the numbers 1 and 47. For instance, a(13) = 47 but (47^2+1)/2 = 1105 is not prime, but 47^2+1 = 2210 => k^2209 == {1, 2, 3, ..., 2208, 2209} mod 2210 for k = {1, 2, ..., 2210}.

Conclusion: the two numbers of this sequence 1, 47 are not in A002731. Are there other numbers?

Examples

			3 is in the sequence because, for q = 3^2 + 1 = 10 we obtain the congruences:
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.
		

Crossrefs

Cf. A002731.

Programs

Maple

with(numtheory):for n from 1  by 2 to 500 do:q:=n^2+1:if type(x,prime)=false then j:=0:for i from 1 to q do: if irem(i^(q-1),q)=i then j:=j+1:else fi:od:if j=q-1 then printf(`%d, `, n):else fi:fi:od:

Mathematica

f[n_] := Module[{q = n^2 + 1}, And @@ Table[PowerMod[k, q - 1, q] == k, {k, q - 1}]]; Select[Range[345], f] (* T. D. Noe, Sep 03 2012 *)