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A147639 Numbers C which generate successive records of the merit function of the ABC conjecture admitting only C which are powers of 2.

%I A147639 #16 Aug 04 2019 07:39:17
%S A147639 4,8,16,32,64,128,1048576
%N A147639 Numbers C which generate successive records of the merit function of the ABC conjecture admitting only C which are powers of 2.
%C A147639 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.
%C A147639 If the ABC conjecture is true this sequence is finite.
%C A147639 The associated numbers B for this case are A147638, the associated A are A147640.
%H A147639 Abderrahmane Nitaj, <a href="https://nitaj.users.lmno.cnrs.fr/abc.html">The ABC conjecture homepage</a>
%e A147639 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.
%p A147639 Digits := 120 : A007947 := proc(n) local f, p; f := ifactors(n)[2] ; mul( op(1, p), p=f) ; end:
%p A147639 L := proc(A, B, C) local rad; rad := A007947(A*B*C) ; evalf(log(C)/log(rad)) ; end:
%p A147639 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
%Y A147639 Cf. A085152, A085153, A147298-A147307, A147638-A147643.
%K A147639 nonn,more
%O A147639 1,1
%A A147639 _Artur Jasinski_, Nov 09 2008
%E A147639 a(2) corrected by _R. J. Mathar_, Aug 28 2009