OFFSET
1,2
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?
EXAMPLE
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.
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 *)
CROSSREFS
KEYWORD
nonn
AUTHOR
Michel Lagneau, Sep 01 2012
STATUS
approved