A320883 3-smooth numbers of the form (ab+1)(ac+1), a > b > c > 0.

96, 288, 3888, 4608, 31104, 69984, 2654208, 2985984, 4478976, 1088391168, 1528823808, 440301256704

Offset: 1

Author

M. F. Hasler, Nov 19 2018

Keywords

• nonn
• fini

Comments

Subsequence of A320884 = 5-smooth terms of A180045, finite according to Corvaja & Zannier.

Can someone prove that a(12) = 440301256704 = (2359*889 + 1)(2359*89 + 1) = 2^26 * 3^8 is the last term?

Links

• P. Corvaja and U. Zannier, On the greatest prime factor of (ab+1)(ac+1), Proceedings of the American Mathematical Society 131 (2003), pp. 1705-1709. See also arXiv:math/0205136 [math.NT], 2002.

Crossrefs

Cf. A180045 = {(ab+1)(ac+1); a > b > c > 0}, A320884 (5-smooth terms of A180045), A003586 (3-smooth numbers).

Programs

Mathematica

(* This is only a recomputation of the existing sequence. *)
(* Max exponents: *) jmax = 26; kmax = 12;
r[j_, k_] := Reduce[a > b > c > 0 && (a b + 1)(a c + 1) == 2^j*3^k , {a, b, c}, Integers];
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 *)

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.
is_A320883(n)={vecmax(factor(n,3)[,1])<4 && is_A180045(n)}

Formula

Intersection of A180045 = {(ab+1)(ac+1); a > b > c > 0} and A003586 (3-smooth numbers).