A147306 - OEIS

%I #11 Aug 04 2019 07:39:10

%S 6,12,18,24,36,48,54,144,384,486,648,2304,3888,5832,279936

%N Numbers C in successive records of the merit function of the ABC conjecture considering only C from A033845.

%C 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 A033845, which avoids some early larger records that would be created by unrestricted C, and leads to a slower increase of the L-values.

%C If the ABC conjecture is true this sequence is finite.

%C The associated numbers B for this case are A147305, the associated A are A147307.

%e (A,B,C) = (1,5,6) defines the first record, L=0.5268.. (A,B,C)=(1,11,12) reaches L=0.5931..

%e (A,B,C) = (1,17,18) reaches L=0.6249.. The first C-number selected from A033845 that does not generate a new record is 72.

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

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

%p isA033845 := proc(n) local f,p; f := ifactors(n)[2] ; for p in f do if not op(1,p) in {2,3} then RETURN(false) ; fi; od: RETURN( (n mod 2 = 0 ) and (n mod 3 = 0 ) ) ; end:

%p crek := -1 : for C from 3 do if isA033845(C) then 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: fi; od: # _R. J. Mathar_, Aug 24 2009

%Y Cf. A085152, A085153, A147298-A147307.

%K nonn,less,more

%O 1,1

%A _Artur Jasinski_, Nov 09 2008

%E Edited by _R. J. Mathar_, Aug 24 2009