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%I #9 Nov 27 2024 17:56:59
%S 1,81,1296,2187,1419857
%N Smaller of consecutive terms b < c of A076467 such that the quality q=log(rad(c))/log(rad((c-b)*b*c)) of the abc-triple c-b,b,c with gcd(c-b,b,c)=1 sets a new record.
%C If it exists, a(6)>5*10^27.
%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Abc_conjecture">abc conjecture</a>
%e Pairs b,c of consecutive
%e terms of A076467
%e A378165
%e c-b b c Quality = log(rad(c))/log(rad((c-b)*b*c))
%e 7, 1, 8, 0.262649535...
%e 44, 81, 125, 0.277532712...
%e 35, 1296, 1331, 0.309605475...
%e 10, 2187, 2197, 0.429917243...
%e 23040, 1419857, 1442897, 0.431260235...
%o (PARI) \\ Uses _M. F. Hasler_'s A076467_vec from A076467
%o a378164_5(upto) = {my(W=A076467_vec(upto), qw=0); for(k=2, #W, my(d=W[k]-W[k-1]); if(gcd([d,W[k],W[k-1]])==1, my(C=factor(W[k])[,1], B=factor(W[k-1])[,1], A=factor(d)[,1], P=vecprod(setunion(setunion(Set(B),Set(C)),Set(A))), q=log(vecprod(C))/log(P)); if(q>qw, print([d,W[k-1],W[k],q]); qw=q)))};
%o a378164_5(10^16)
%Y A378165 gives the corresponding values of c-b.
%Y Cf. A007947 (rad), A076467, A377933, A377934, A378166, A378167.
%K nonn,hard,more
%O 1,2
%A _Hugo Pfoertner_, Nov 18 2024