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.

11, 27, 243, 25, 5041, 9747, 1681, 67, 2875, 361, 2187, 841, 16807, 19683, 29, 50653, 361, 121, 173, 513, 125, 28561, 1369, 78125, 78125, 2197, 2187, 243, 125, 95, 3479, 15625, 279841, 83521, 337, 847, 62083, 137781, 378125, 40817, 484183, 343, 8281, 89167, 15625

Offset: 1

Author

Frank M Jackson, Sep 11 2024

Keywords

• nonn

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 c can be found at A376143 and the sequence indicating whether a is 1 or -1 can be found at A376149.

Examples

			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.
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.
		

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, A376149.

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}}]; Part[#,2]&/@SortBy[lst,{#[[2]]&,#[[3]]&}]

Extensions

More terms from David A. Corneth, Sep 18 2024