A060121 First solution mod p of x^3 = 2 for primes p such that only one solution exists.

0, 2, 3, 7, 8, 16, 26, 5, 21, 18, 38, 49, 50, 16, 26, 6, 81, 54, 98, 70, 157, 161, 58, 147, 21, 86, 92, 197, 50, 249, 137, 184, 119, 45, 45, 261, 198, 61, 176, 143, 51, 103, 221, 72, 11, 219, 35, 86, 385, 384, 141, 143, 225, 92, 245, 533, 557, 473, 170, 375, 516

Offset: 1

Author

Klaus Brockhaus, Mar 02 2001

Keywords

• nonn

Comments

Solutions mod p are represented by integers from 0 to p-1. For i > 1, i is a solution mod p of x^3 = 2 iff p is a prime factor of i^3-2 and p > i (cf. comment to A059940). i^3-2 has at most two prime factors > i. Hence i is a solution mod p of x^3 = 2 for at most two different p and therefore no integer occurs more than twice in this sequence. There are integers which do occur twice, e.g. 16, 21, 26 (cf. A060914). Moreover, no integer occurs more than twice in A060121, A060122, A060123 and A060124 taken together.

Examples

			a(9) = 21, since 47 is the ninth term of A045309 and 21 is the only solution mod 47 of x^3 = 2.
		

Links

• Robert Israel, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A040028, A045309, A059940, A060914, A060122, A060123, A060124.

Programs

Maple

Res:=0,2: count:= 2: p:= 3:
while count < 100 do
p:= nextprime(p);
   if p mod 3 = 2 then
    count:= count+1;
    Res:= Res, numtheory:-mroot(2,3,p);
fi
od:
Res; # Robert Israel, Sep 12 2018

Mathematica

terms = 100;
A045309 = Select[Prime[Range[2 terms]], Mod[#, 3] != 1&];
a[n_] := PowerMod[2, 1/3, A045309[[n]]];
Array[a, terms] (* Jean-François Alcover, Feb 27 2019 *)

Formula

a(n) = first (only) solution mod p of x^3 = 2, where p is the n-th prime such that x^3 = 2 has only one solution mod p, i.e. p is the n-th term of A045309.