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A147306
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Numbers C in successive records of the merit function of the ABC conjecture considering only C from A033845.
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7
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6, 12, 18, 24, 36, 48, 54, 144, 384, 486, 648, 2304, 3888, 5832, 279936
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OFFSET
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1,1
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COMMENTS
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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.
If the ABC conjecture is true this sequence is finite.
The associated numbers B for this case are A147305, the associated A are A147307.
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LINKS
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EXAMPLE
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(A,B,C) = (1,5,6) defines the first record, L=0.5268.. (A,B,C)=(1,11,12) reaches L=0.5931..
(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.
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MAPLE
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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:
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:
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
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CROSSREFS
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KEYWORD
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nonn,less,more
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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