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A342390 Numbers k such that the ring of integers of Q(2^(1/k)) is not Z[2^(1/k)].

%I A342390 #21 Mar 20 2022 02:20:10
%S A342390 1093,2186,3279,3511,4372,5465,6558,7022,7651,8744,9837,10533,10930,
%T A342390 12023,13116,14044,14209,15302,16395,17488,17555,18581,19674,20767,
%U A342390 21066,21860,22953,24046,24577,25139,26232,27325,28088,28418,29511,30604,31599,31697,32790
%N A342390 Numbers k such that the ring of integers of Q(2^(1/k)) is not Z[2^(1/k)].
%C A342390 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 A342390 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 A342390 Here a = 2, and the only known Wieferich primes of base 2 (A001220) are 1093, 3511 are no more below 4.97*10^17. So all known terms are multiples of either 1093 or 3511 (or both).
%H A342390 Jianing Song, <a href="/A342390/b342390.txt">Table of n, a(n) for n = 1..11995</a> (all terms up to 10^7).
%H A342390 Keith Conrad, <a href="https://kconrad.math.uconn.edu/blurbs/gradnumthy/integersradical.pdf">The ring of integers in a radical extension</a>
%e A342390 2^364 == 1 (mod 1093^2), so x = (1 + 2^(364/1093) + 2^(2*364/1093) + ... + 2^(1092*364/1093))/1093 is an algebraic integer not in Z[2^(1/1093)].
%o A342390 (PARI) Wieferich_up_to_n(lim) = my(v=[]); forprime(p=2, lim, if(Mod(2,p^2)^(p-1)==1, v=concat(v,p))); v
%o A342390 Up_to_n(lim) = my(pv=Wieferich_up_to_n(lim), list=[]); for(i=1, #pv, my(p=pv[i]); for(k=1, lim\p, list=concat(list, k*p))); list=Set(list); list \\ corrected by _Jianing Song_, Mar 19 2022
%o A342390 (PARI) Up_to_n(lim) = my(list=[]); for(k=1, lim\1093, list=concat(list, 1093*k)); for(k=1, lim\3511, list=concat(list, 3511*k)); list=Set(list); list \\ valid up to 4.97*10^17, corrected by _Jianing Song_, Mar 19 2022
%Y A342390 Cf. A001220, A342391.
%K A342390 nonn
%O A342390 1,1
%A A342390 _Jianing Song_, Mar 10 2021