A147642 Numbers C which generate successive records of the merit function of the ABC conjecture admitting only C which are powers of 23.

23, 529, 12167, 6436343

Offset: 1

Author

Artur Jasinski, Nov 09 2008

Keywords

• nonn
• more

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 L-values.

For associated B for this case see A147641, for associated A see A147643.

Examples

			C= 23 is the first candidate (and therefore by definition a record). Scanning the pairs (A,B) for this C we have L-values 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 L-value 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 L-value for C=529 is larger than the L-value 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 C-value of 12167 is again a record-holder.
		

Links

• OEIS Index entries for sequences related to the abc conjecture.

Crossrefs

Cf. A085152, A085153, A147298 - A147307, A147638 - A147643.