1,1

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 sequence A000079, which avoids some early larger records that would be created by unrestricted C, and leads to a slower increase of the L-values.

If the ABC conjecture is true this sequence is finite.

The associated numbers B for this case are A147638, the associated A are A147640.

Abderrahmane Nitaj, The ABC conjecture homepage

The case C=2 does not create a valid (A,B,C) triple, so C=4 is the first case, which sets a first record L=0.7737 with (A,B,C)=(1,3,4). The next admitted case, C=8, sets a new record L=0.7879 with (A,B,C)=(1,7,8), and so do (A,B,C)=(1,15,16) with L=0.8151. For C=32, we consider the largest L possible for A<B<C, which is (A,B,C)=(5,27,32) with L=1.0189. The value L=0.839 from (A,B,C)=(1,31,32) at the same C is smaller and discarded.

Digits := 120 : A007947 := proc(n) local f, p; f := ifactors(n)[2] ; mul( op(1, p), p=f) ; end:

L := proc(A, B, C) local rad; rad := A007947(A*B*C) ; evalf(log(C)/log(rad)) ; end:

crek := -1 : for x from 2 do C := 2^x ; for A from 1 to C/2 do B := C-A ; if gcd(A, B) = 1 then l := L(A, B, C) ; if l > crek then print(C) ; crek := l ; fi; fi; od: od: # R. J. Mathar, Aug 28 2009

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

Sequence in context: A186949 A020707 A151821 * A049934 A328634 A089890

Adjacent sequences: A147636 A147637 A147638 * A147640 A147641 A147642

nonn,more

Artur Jasinski, Nov 09 2008

a(2) corrected by R. J. Mathar, Aug 28 2009

approved