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

1,1

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.

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.

Example

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 in 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 in the form a+b+c = d we have 1 + 5^2 + 109^2 = 3^5 * 7^2.

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]]&}]

Crossrefs

Cf. A007947, A216323, A376143, A376149.

Sequence in context: A044072 A044453 A212776 * A362528 A137019 A122929

Adjacent sequences: A376141 A376142 A376143 * A376145 A376146 A376147

Keyword

nonn

Author

Frank M Jackson, Sep 11 2024

Extensions

More terms from David A. Corneth, Sep 18 2024

Status

approved