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%I #14 Dec 07 2018 16:47:47
%S 96,288,3888,4608,31104,69984,2654208,2985984,4478976,1088391168,
%T 1528823808,440301256704
%N 3-smooth numbers of the form (ab+1)(ac+1), a > b > c > 0.
%C Subsequence of A320884 = 5-smooth terms of A180045, finite according to Corvaja & Zannier.
%C Can someone prove that a(12) = 440301256704 = (2359*889 + 1)(2359*89 + 1) = 2^26 * 3^8 is the last term?
%H P. Corvaja and U. Zannier, <a href="https://doi.org/10.1090/S0002-9939-02-06771-0">On the greatest prime factor of (ab+1)(ac+1)</a>, Proceedings of the American Mathematical Society 131 (2003), pp. 1705-1709. See also <a href="https://arxiv.org/abs/math/0205136">arXiv:math/0205136 [math.NT]</a>, 2002.
%F Intersection of A180045 = {(ab+1)(ac+1); a > b > c > 0} and A003586 (3-smooth numbers).
%t (* This is only a recomputation of the existing sequence. *)
%t (* Max exponents: *) jmax = 26; kmax = 12;
%t r[j_, k_] := Reduce[a > b > c > 0 && (a b + 1)(a c + 1) == 2^j*3^k , {a, b, c}, Integers];
%t Reap[Do[rr = r[j, k]; If[rr =!= False, Print[{j, k, 2^j*3^k}]; Sow[2^j*3^k]], {j, 1, jmax}, {k, 1, kmax}]][[2, 1]] // Union (* _Jean-François Alcover_, Dec 05 2018 *)
%o (PARI) A320883(LIM=35,S=[])={for(m=1,LIM, for(k=0,m, is_A180045(3^k<<(m-k))&& S=setunion(S,[3^k<<(m-k)])));S} \\ Gives all terms up to 2^LIM and possibly some larger terms up to 3^LIM.
%o is_A320883(n)={vecmax(factor(n,3)[,1])<4 && is_A180045(n)}
%Y Cf. A180045 = {(ab+1)(ac+1); a > b > c > 0}, A320884 (5-smooth terms of A180045), A003586 (3-smooth numbers).
%K nonn,fini
%O 1,1
%A _M. F. Hasler_, Nov 19 2018
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