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A376149 Numbers a = +/- 1 such that a + b + c = d are abcd quadruples in the "abcd-conjecture" with a < b < c < d, all |a|, b, c, d are pairwise coprime, the quality q of the quadruple has q > 1, term b = A376144(n) and term c = A376143(n). Quadruples are sorted by c then b.

%I A376149 #23 Apr 02 2025 04:12:23
%S A376149 -1,-1,1,1,1,1,1,1,1,1,1,1,1,-1,-1,1,1,1,-1,1,-1,-1,-1,1,1,-1,1,1,-1,
%T A376149 -1,-1,1,1,1,1,1,1,1,-1,-1,1,1,1,1,1,-1,1,1,1,1,-1,-1,1,1,1,-1,-1,1,1,
%U A376149 1,1,-1,-1,1,1,-1,1,1,-1,1,-1,1,1,1,-1,-1,1,1,1,1
%N A376149 Numbers a = +/- 1 such that a + b + c = d are abcd quadruples in the "abcd-conjecture" with a < b < c < d, all |a|, b, c, d are pairwise coprime, the quality q of the quadruple has q > 1, term b = A376144(n) and term c = A376143(n). Quadruples are sorted by c then b.
%C A376149 An abcd quadruple is defined as (a, b, c, d) with a+b+c+d = 0, all |a|, |b|, |c|, |d| are pairwise coprime, and radical of a*b*c*d, rad(|a|*|b|*|c|*|d|) < max (|a|, |b|, |c|, |d|).
%C A376149 The quality of an abcd quadruple is q = log(max(|a|,|b|,|c|,|d|))/log(rad(|a|*|b|*|c|*|d|)).
%C A376149 This sequence considers quadruples of the form a = +/- 1 and a+b+c = d with a < b < c < d.
%C A376149 Corresponding numbers b can be found at A376144 and corresponding numbers c can be found at A376143.
%H A376149 David A. Corneth, <a href="/A376149/b376149.txt">Table of n, a(n) for n = 1..161</a>
%H A376149 C. F. W. Ramaekers, <a href="https://www.win.tue.nl/~bdeweger/downloads/CoenRamaekersBachelorThesis.pdf">The abc-Conjecture and the n-conjecture</a>, Eindhoven University of Technology Nov 12, 2009.
%e A376149 a(2) = 27 because the second occurrence of an abcd quadruple with a = +/- 1 is (-1, 27, 2375, 2401) with a = -1. As factors of the form a+d = b+c we have 1 + 7^4 = 3^3 + 5^3 * 19.
%e A376149 a(4) = 25 because the fourth occurrence of an abcd quadruple with a = +/- 1 is (1, 25, 11881, 11907) with a = 1. As factors of the form a+b+c = d we have 1 + 5^2 + 109^2 = 3^5 * 7^2.
%t A376149 Rad[n_] := Module[{lst=FactorInteger[n]}, Times@@(First/@lst)]; lst={}; Do[Do[If[d=b+c+a; AllTrue[{{Abs[a],b},{Abs[a],c},{Abs[a],d},{b,c},{b,d},{c,d}}, Apply[CoprimeQ]]&&d>Rad[Abs[a]*b*c*d], AppendTo[lst,{a,b,c}]], {c, 3, 3000}, {b, 2, c}], {a, {-1, 1}}]; First/@SortBy[lst,{#[[2]]&,#[[3]]&}]
%Y A376149 Cf. A007947, A216323, A376143, A376144.
%K A376149 sign
%O A376149 1,1
%A A376149 _Frank M Jackson_, Sep 12 2024
%E A376149 More terms from _David A. Corneth_, Sep 18 2024