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A320885 7-smooth but not 5-smooth numbers of the form (ab+1)(ac+1), a > b > c > 0.

%I A320885 #18 Dec 07 2018 12:27:26
%S A320885 28,126,175,280,336,378,441,560,630,672,1225,1470,1680,1701,1792,2016,
%T A320885 2520,2835,3136,3969,4200,5250,5600,6860,7840,7875,8400,8960,9072,
%U A320885 9408,11025,11340,12096,13125,15120,17640,19845,20160,21000,23520,24696,27440,30625,32928,35000
%N A320885 7-smooth but not 5-smooth numbers of the form (ab+1)(ac+1), a > b > c > 0.
%C A320885 Corvaja & Zannier show that there are only finitely many p-smooth terms in A180045, for any prime p. This sequences lists these terms for p = 7 without those for p = 5 (A320884), and is therefore finite.
%H A320885 David A. Corneth, <a href="/A320885/b320885.txt">Table of n, a(n) for n = 1..910</a> (first 671 terms by Maximilian Hasler; terms < 10^35, the largest of which is ~2.6*10^31)
%H A320885 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 A320885 Intersection of A080194 (gpf(n) = 7) and A180045 ((ab+1)(ac+1)).
%t A320885 Reap[For[k = 7, k <= 35000, k = k+7, If[FactorInteger[k][[-1, 1]] == 7, If[ Reduce[k == (a b + 1)(a c + 1) && a > b > c > 0, {a, b, c}, Integers] =!= False, Print[k]; Sow[k]]]]][[2, 1]] (* _Jean-François Alcover_, Dec 07 2018 *)
%o A320885 (PARI) is_A320885(n)={vecmax(factor(n,7)[,1])==7 && is_A180045(n)}
%o A320885 A320885=select( is_A180045, A080194_list(1e20)) \\ Only initial terms, not the complete sequence. For more efficiency, use is_A180045 or a dedicated implementation inside the nested loops in A080194_list().
%Y A320885 Cf. A080194 (greatest prime factor = 7).
%Y A320885 Cf. A180045 (numbers (ab+1)(ac+1), a>b>c>0), A320883 (subsequence of 3-smooth terms), A320884 (subsequence of 5-smooth terms).
%K A320885 nonn,fini
%O A320885 1,1
%A A320885 _M. F. Hasler_, Nov 21 2018