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 A379524 #34 Feb 07 2025 17:21:06 %S A379524 32009,214712,710652,8127208,180527768 %N A379524 Minimal discriminants d of real quadratic number fields K = Q(sqrt(d)), d > 0, with elementary bicyclic 3-class group Cl_3(K)=(3,3) and second 3-class group M=Gal(F_3^2(K)/K) of assigned coclass cc(M)=1,2,3,4,... %C A379524 The coclass cc(M) for the field K with discriminant d=a(n) is n, and for each field K with discriminant d < a(n), the coclass cc(M) is less than n. %C A379524 The Magma program "RealCoClass.m" in the Links is independent of the data file "ipad_freq_real" by M. R. Bush. It computes the first five terms, 180527768 inclusively, in precisely 14 days of CPU time on an Intel Core i7 4790 quadcore processor with clock rate 4.0 GHz. %D A379524 M. R. Bush, ipad_freq_real, file with two lists, disclist and ipadlist, containing all IPADs of real quadratic fields K with 3-class group of rank 2 and discriminant d < 10^9, Washington and Lee Univ. Lexington, Virginia, 2015. %H A379524 Daniel Constantin Mayer, <a href="http://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 (2013), 401-456. %H A379524 Daniel Constantin Mayer, <a href="http://arxiv.org/abs/1403.3839">Principalization algorithm via class group structure</a>, arXiv:1403.3839 [math.NT], 2014; J. Théor. Nombres Bordeaux 26 (2014), 415-464. %H A379524 Daniel Constantin 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 A379524 Daniel Constantin Mayer, <a href="/A379524/a379524.m.txt">M. R. Bush: data file ipad_freq_real</a> %H A379524 Daniel Constantin Mayer, <a href="/A379524/a379524_1.m.txt">Program "SiftRealIPADs.m" which extracts minimal discriminants for assigned IPADs from the file ipad_freq_real and arranges them in the table "IpadFreqReal"</a> %H A379524 Daniel Constantin Mayer, <a href="/A379524/a379524.txt">"IpadFreqReal": table of minimal discriminants for assigned IPADs</a> %H A379524 Daniel Constantin Mayer, <a href="/A379524/a379524_2.m.txt">Magma program "RealCoClass.m" with endless loop</a> %F A379524 According to Theorem 3.12 on page 435 of "The distribution of second p-class groups on coclass graphs", the coclass of the group M is given by cc(M)+1=log_3(h_3(L_2)), where h_3(L_2) is the second largest 3-class number among the four unramified cyclic cubic extensions L_1,..,L_4 of the quadratic field K. Thus, cc(M) is determined uniquely by the IPAD of K. %e A379524 We have cc(M)=1 for d=32009, cc(M)=2 for d=214712, cc(M)=3 for d=710652, cc(M)=4 for d=8127208, cc(M)=5 for d=180527768. The Magma script "SiftRealIPADs.m" produces a table "IpadFreqReal" of minimal discriminants for each IPAD from the file ipad_freq_real. This table admits the determination of the term a(n) of the sequence A379524. For instance: According to the FORMULA, the table contains three candidates for a(4) with cc(M)=4 and thus cc(M)+1=5=log_3(3^5)=log_3(#[9,27])=log_3(h_3(L_2)) with the second largest 3-class number h_3(L_2) in the IPAD. They are 8321505 and 8491713 and 8127208. Thus the minimal discriminant is a(4)=8127208. %o A379524 (Magma) // See Links section. %Y A379524 Cf. A269318, A269319 (supersequences). %K A379524 nonn,hard,more %O A379524 1,1 %A A379524 _Daniel Constantin Mayer_, Dec 24 2024