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A346420 a(n) is twice the coefficient of the radical part in the fundamental unit of Q(sqrt(A000037(n))) where A000037 lists the nonsquare numbers (Version 2).

%I A346420 #13 Feb 16 2025 08:34:02
%S A346420 2,2,1,4,6,2,2,6,2,1,8,2,2,2,78,1,1,84,10,4,2,2,6,1,4,546,2,8,12,2,2,
%T A346420 12,8,2,10,4,1062,6,1,7176,14,2,2,14,1,1,4,24,8,40,26,138,2,5,16,6,2,
%U A346420 16,11934,2,3,60,826,2,250,10,2,78,1,12,18,1,2,18,1,1,2244,6,84
%N A346420 a(n) is twice the coefficient of the radical part in the fundamental unit of Q(sqrt(A000037(n))) where A000037 lists the nonsquare numbers (Version 2).
%C A346420 The radical part is actually sqrt(A007913(A000037(n))) where A007913(m) is the squarefree part of m. - _Michel Marcus_, Jun 26 2020
%C A346420 How does this sequence differ from A048942? The definitions of both sequences are identical, but the second comment in A048942 states the terms differ from n = 14 onwards. - _Felix Fröhlich_, Jun 16 2022
%H A346420 S. R. Finch, <a href="http://www.people.fas.harvard.edu/~sfinch/">Class number theory</a>
%H A346420 Steven R. Finch, <a href="/A000924/a000924.pdf">Class number theory</a> [Cached copy, with permission of the author]
%H A346420 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/FundamentalUnit.html">Fundamental Unit.</a>
%o A346420 (PARI) f(n) = {if (issquare(n), return (0)); if (!issquarefree(n), m = core(n), m = n); my(u = abs(2*polcoeff(lift(bnfinit(x^2-m, 1).fu[1]), 0))); if (u^2==1, return (1)); if (u^2==4, return (sqrtint((u^2+4)/m));); if (u^2 < 4, return((u^2+4)/n)); my(v2 = [(u^2-4)/m, (u^2+4)/m]); sqrtint(vecmin(select(x->denominator(x)==1, v2)));}
%o A346420 lista(nn) = apply(f, select(x->!issquare(x), [1..nn])); \\ _Michel Marcus_, Jun 25 2020; corrected Jun 16 2022
%Y A346420 Cf. A000037, A007913, A346419, A048941, A048942.
%K A346420 nonn
%O A346420 1,1
%A A346420 _Sean A. Irvine_