A060121 - OEIS

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

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OFFSET

1,2

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.

LINKS

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

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.

EXAMPLE

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

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);

od:

Res; # Robert Israel, Sep 12 2018

MATHEMATICA