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.

-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, -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, 1, -1, -1, 1, 1, -1, 1, 1, -1, 1, -1, 1, 1, 1, -1, -1, 1, 1, 1, 1

Offset: 1

Author

Frank M Jackson, Sep 12 2024

Keywords

• sign

Comments

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|).

The quality of an abcd quadruple is q = log(max(|a|,|b|,|c|,|d|))/log(rad(|a|*|b|*|c|*|d|)).

This sequence considers quadruples of the form a = +/- 1 and a+b+c = d with a < b < c < d.

Corresponding numbers b can be found at A376144 and corresponding numbers c can be found at A376143.

Examples

			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.
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.
		

Links

• David A. Corneth, Table of n, a(n) for n = 1..161
• C. F. W. Ramaekers, The abc-Conjecture and the n-conjecture, Eindhoven University of Technology Nov 12, 2009.

Crossrefs

Cf. A007947, A216323, A376143, A376144.

Programs

Mathematica

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]]&}]

Extensions

More terms from David A. Corneth, Sep 18 2024