This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A247693 #23 May 27 2025 10:08:16
%S A247693 16627,262744,4776071,40059363
%N A247693 Minimal absolute discriminants a(n) of complex quadratic fields with 3-class group of type (3,3), 3-principalization type E.14 (3122), and second 3-class group G of odd nilpotency class cl(G)=2(n+2)+1.
%C A247693 The 3-principalization type (transfer kernel type, TKT) E.14 (3122) is not a permutation, contains a 3-cycle, and has no fixed points.
%C A247693 The nilpotency condition cl(G)=2n+5 for the second 3-class group is equivalent to a transfer target type, TTT (called IPAD by Boston, Bush and Hajir) of the shape [(3^{n+2},3^{n+3}),(3,3,3),(3,9)^2].
%C A247693 The second 3-class group G is one of two vertices of depth 1 on the coclass tree with root SmallGroup(243,6) contained in the coclass graph G(3,2).
%C A247693 All these fields possess a Hilbert 3-class field tower of exact length 3.
%C A247693 A247693 is an extremely sparse subsequence of A242878 and it is exceedingly hard to compute a(n) for n>0.
%H A247693 N. Boston, M. R. Bush, and F. Hajir, <a href="http://arxiv.org/abs/1111.4679">Heuristics for p-class towers of imaginary quadratic fields</a>, Math. Ann. (2013), Preprint: arXiv:1111.4679v1 [math.NT], 2011.
%H A247693 M. R. Bush and D. C. Mayer, <a href="http://arxiv.org/abs/1312.0251">3-class field towers of exact length 3</a>, J. Number Theory (2014), Preprint: arXiv:1312.0251v1 [math.NT], 2013.
%H A247693 D. C. Mayer, <a href="https://arxiv.org/abs/1403.3899">The second p-class group of a number field</a>, arXiv:1403.3899 [math.NT], 2014; Int. J. Number Theory 8 (2012), no. 2, 471-505.
%H A247693 D. C. Mayer, <a href="https://arxiv.org/abs/1403.3896">Transfers of metabelian p-groups</a>, arXiv:1403.3896 [math.GR], 2014; Monatsh. Math. 166 (3-4) (2012), 467-495.
%H A247693 D. C. Mayer, <a href="https://arxiv.org/abs/1403.3833">The distribution of second p-class groups on coclass graphs</a>, arXiv:1403.3833 [math.NT], 2014; J. Théor. Nombres Bordeaux 25 (2) (2013), 401-456.
%H A247693 D. C. Mayer, <a href="http://arxiv.org/abs/1403.3839">Principalization algorithm via class group structure</a>, J. Théor. Nombres Bordeaux (2014), Preprint: arXiv:1403.3839v1 [math.NT], 2014.
%H A247693 D. C. Mayer, <a href="https://arxiv.org/abs/1504.00851">Periodic sequences of p-class tower groups</a>, arXiv:1504.00851 [math.NT], 2015.
%H A247693 Wikipedia, <a href="https://en.wikipedia.org/wiki/Artin_transfer_(group_theory)#Example">Artin transfer (group theory), Table 2</a>
%e A247693 For a(0)=16627, we have the ground state of TKT E.14 with TTT [(9,27),(3,3,3),(3,9)^2] and cl(G)=5.
%e A247693 For a(1)=262744, we have the first excited state of TKT E.14 with TTT [(27,81),(3,3,3),(3,9)^2] and cl(G)=7.
%e A247693 a(0) and a(1) are due to D. C. Mayer (2012).
%e A247693 a(2) and a(3) are due to N. Boston, M. R. Bush and F. Hajir (2013).
%Y A247693 Cf. A242862, A242863, A242878 (supersequences), A247692, A247694, A247695, A247696, A247697 (disjoint sequences).
%K A247693 hard,more,nonn
%O A247693 0,1
%A A247693 _Daniel Constantin Mayer_, Sep 28 2014