
COMMENTS

In a variant of the ABC conjecture (see A120498) we look at triples (A,B,C) restricted to A+B=C, gcd(A,B)=1, and at the merit function L(A,B,C)=log(C)/log(rad(A*B*C)), where rad() is the squarefree kernel A007947, as usual. Watching for records in L() as C runs through the integers generates A147302. In this sequence here, we admit only the C of the form 23^x, see A009967, which avoids some early larger records that would be created by unrestricted C, and leads to a slower increase of the Lvalues.
For associated B for this case see A147641, for associated A see A147643.


EXAMPLE

C= 23 is the first candidate (and therefore by definition a record). Scanning the pairs (A,B) for this C we have Lvalues of L(1,22,23) = 0.5035, L(2,21,23) = 0.456, ... L(6,17,23) = 0.404, L(7,16,23) = 0.542 ,... L(11,12,23) = 0.428. The largest Lvalue stems from (A=7,B=16) which means the representative triple of the first record is (A,B,C) = (7,16,23).
C= 23^2= 529 is the next candidate. Scanning again all (A,B) values subject to the constraints we achieve L(17,512,529) = 0.941... (Smaller ones like L(81,448,529) = 0.9123... are discarded). Since the Lvalue for C=529 is larger than the Lvalue for C=23, the next record is C=529 with representatives (A,B,C)= (17,512,529).
The third candidate is C= 23^3= 12167. This generates a maximum of L(162,12005,12167) = 1.1089... (smaller values like L(17,12150,12167) = 1.0039.. discarded) which is again larger than the maximum of the previous record (which was 0.941..) So the Cvalue of 12167 is again a recordholder.
