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A376143 Numbers c 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 b = A376144(n) (with repetitions and sorted by c then b).

%I A376143 #38 Apr 26 2025 20:13:25
%S A376143 2187,2375,2401,11881,14641,14641,15125,16807,16807,19321,22201,28561,
%T A376143 28561,42875,50625,72171,77763,78003,83349,83521,85169,101761,116281,
%U A376143 121801,127253,128125,146689,152881,177023,177147,177147,249001,303601,369603,390625,390625
%N A376143 Numbers c 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 b = A376144(n) (with repetitions and sorted by c then b).
%C A376143 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 A376143 The quality of an abcd quadruple is q = log(max(|a|,|b|,|c|,|d|))/log(rad(|a|*|b|*|c|*|d|)).
%C A376143 This sequence considers quadruples of the form a = +/- 1 and a+b+c = d with a < b < c < d.
%C A376143 Corresponding numbers for b can be found at A376144 and the sequence indicating whether a is 1 or -1 can be found at A376149.
%C A376143 From _David A. Corneth_, Sep 17 2024: (Start)
%C A376143 All of a, b, c, d in a valid quadruple are odd. Proof: As a + b + c = d, a + b + c + d = 2d is even. Hence an even number of numbers in {a, b, c, d} is odd and as there is an even number of numbers in {a, b, c, d} consequently an even number of them is even.
%C A376143 If more than 0 of them is even then at least two of them are even and not all of {a, b, c, d} are pairwise coprime. Therefore exactly 0 of them are even i.e. they are all odd. (End)
%H A376143 David A. Corneth, <a href="/A376143/b376143.txt">Table of n, a(n) for n = 1..161</a>
%H A376143 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.
%H A376143 David A. Corneth, <a href="/A376143/a376143.gp.txt">quadruples (a, b, c, d) with b + c <= 10^7</a>
%e A376143 a(2) = 2375 because the second occurrence of these abcd quadruples with a = +/- 1 is (-1, 27, 2375, 2401) with c = 2375. As prime factors of the form a+d = b+c, we have 1 + 7^4 = 3^3 + 5^3 * 19.
%e A376143 a(4) = 11881 because the fourth occurrence of these abcd quadruples with a = +/- 1 is (1, 25, 11881, 11907) with c = 11881. As prime factors of the form a+b+c = d, we have 1 + 5^2 + 109^2 = 3^5 * 7^2.
%e A376143 From _David A. Corneth_, Sep 16 2024: (Start)
%e A376143 Suppose we look for terms <= 2500. We could iterate over squarefree numbers <= 2500. One of the numbers we will see is 285.
%e A376143 List its prime factors; {3,5,19}.
%e A376143 Make all pairs of sets of distinct prime factors.
%e A376143 These are {{{3}, {5, 19}}, {{5}, {3, 19}}, {{19}, {3,5}}}. One of the pairs we will check is {{3}, {5, 19}}. For each set in it, list all numbers of the form 3^k <= 2500 and 5^m*19^t <= 2500 with k, m, t >= 1. These are (3, 9, 27, 81, 243, 729, 2187) and (95, 475, 1805, 2375). Check all pairs from the Cartesian product of these numbers. One pair is (27, 2375).
%e A376143 The smallest is a candidate for b, the largest for c. See if either (or both) of candidate values a in {-1, 1} give a quadruple (a, b, c, d) (where d = a + b + c meeting the quality requirements and pairwise coprimeness (and d <= 2500)). For this example we find the quadruple (-1, 27, 2375, 2401). (End)
%t A376143 Rad[n_] := Module[{lst=FactorInteger[n]}, Times@@(First/@lst)]; lst={}; Do[Do[If[d=b+c+a; AllTrue[{{Abs[a],b},{a,c},{a,d},{b,c},{b,d},{c,d}}, Apply[CoprimeQ]]&&d>Rad[Abs[a]*b*c*d], AppendTo[lst,{a,b,c}]], {c, 1, 3000}, {b, 2, c}], {a, {-1, 1}}]; Last/@SortBy[lst,{#[[2]]&,#[[3]]&}]
%Y A376143 Cf. A007947, A216323, A376144, A376149.
%K A376143 nonn
%O A376143 1,1
%A A376143 _Frank M Jackson_, Sep 11 2024
%E A376143 a(15) corrected by _David A. Corneth_, Sep 16 2024