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A376144 Positive numbers b 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 a = +/- 1 = A376149(n) and term c = A376143(n). Quadruples are sorted by c then b.

%I A376144 #25 Apr 02 2025 04:11:57
%S A376144 11,27,243,25,5041,9747,1681,67,2875,361,2187,841,16807,19683,29,
%T A376144 50653,361,121,173,513,125,28561,1369,78125,78125,2197,2187,243,125,
%U A376144 95,3479,15625,279841,83521,337,847,62083,137781,378125,40817,484183,343,8281,89167,15625
%N A376144 Positive numbers b 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 a = +/- 1 = A376149(n) and term c = A376143(n). Quadruples are sorted by c then b.
%C A376144 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 A376144 The quality of an abcd quadruple is q = log(max(|a|,|b|,|c|,|d|))/log(rad(|a|*|b|*|c|*|d|)).
%C A376144 This sequence considers quadruples of the form a = +/- 1 and a+b+c = d with a < b < c < d.
%C A376144 Corresponding numbers c can be found at A376143 and the sequence indicating whether a is 1 or -1 can be found at A376149.
%H A376144 David A. Corneth, <a href="/A376144/b376144.txt">Table of n, a(n) for n = 1..161</a>
%H A376144 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 A376144 a(2) = 27 because the second occurrence of an abcd quadruple with a = +/- 1 is (-1, 27, 2375, 2401) with b = 27. As prime factors of the form a+d = b+c we have 1 + 7^4 = 3^3 + 5^3 * 19.
%e A376144 a(4) = 25 because the fourth occurrence of an abcd quadruple with a = +/- 1 is (1, 25, 11881, 11907) with b = 25. As prime factors of the form a+b+c = d we have 1 + 5^2 + 109^2 = 3^5 * 7^2.
%t A376144 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}}]; Part[#,2]&/@SortBy[lst,{#[[2]]&,#[[3]]&}]
%Y A376144 Cf. A007947, A216323, A376143, A376149.
%K A376144 nonn
%O A376144 1,1
%A A376144 _Frank M Jackson_, Sep 11 2024
%E A376144 More terms from _David A. Corneth_, Sep 18 2024