A342392 - OEIS

%I #19 Mar 11 2021 04:48:07

%S 2,3,5,6,7,11,13,14,15,21,22,23,29,30,31,33,34,38,39,41,42,43,47,51,

%T 57,58,59,61,65,66,67,69,70,74,77,78,79,83,85,86,87,93,94,95,97,101,

%U 102,103,105,106,110,111,113,114,115,119,122,123,129,130,131,133

%N Cubefree numbers a > 1 such that the ring of integers of Q(a^(1/3)) is Z[a^(1/3)].

%C Squarefree numbers not congruent to 1 or 8 modulo 9.

%C For k > 1, a != 1 being a squarefree number (a != -1 unless k is a power of 2), then the ring of integers of Q(a^(1/k)) is Z[a^(1/k)] if and only if: for every p dividing k, we have a^(p-1) !== 1 (mod p^2). In other words, O_Q(a^(1/k)) = Z[a^(1/k)] if and only if none of the prime factors of k is a Wieferich prime of base a. See Theorem 5.3 of the paper of Keith Conrad.

%C In general, if a^d == 1 (mod p^2) for some d|(p-1), then it is easy to show that x = (1 + a^(d/p) + a^(2*d/p) + ... + a^((p-1)*d/p))/p is an algebraic integer not in Z[a^(1/p)].

%C The asymptotic density of this sequence is 9/(2*Pi^2) = 0.455945... (A088245). - _Amiram Eldar_, Mar 11 2021

%H Jianing Song, <a href="/A342392/b342392.txt">Table of n, a(n) for n = 1..15000</a>

%H Keith Conrad, <a href="https://kconrad.math.uconn.edu/blurbs/gradnumthy/integersradical.pdf">The ring of integers in a radical extension</a>

%e The ring of integers of Q(2^(1/3)) is {a + b*2^(1/3) + c*4^(1/3): a,b,c in Z} = Z[2^(1/3)], so 2 is a term.

%o (PARI) isA342392(n) = (n^2%9!=1) && issquarefree(n)

%Y Cf. A004709 (cubefree numbers), A088245, A342393.

%K nonn,easy

%O 1,1

%A _Jianing Song_, Mar 10 2021