A320884 5-smooth numbers of the form (ab+1)(ac+1), a > b > c > 0.

45, 96, 120, 225, 288, 540, 640, 1080, 1200, 1920, 2160, 3888, 4000, 4500, 4608, 5760, 6480, 7200, 8640, 9600, 10935, 16875, 18225, 25000, 25600, 27000, 28800, 30720, 31104, 38400, 46080, 48600, 69984, 75000, 81000, 91125, 97200, 102400, 112500, 115200, 164025, 184320

Offset: 1

Author

M. F. Hasler, Nov 19 2018

Keywords

• nonn
• fini

Comments

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 = 5, and is therefore finite.

Can someone prove that a(163) = 3327916660110655488000000000 = (16775191*16038089 + 1)(16775191*737369 + 1) = 2^42 * 3^18 * 5^9 is the last term? - M. F. Hasler, Nov 19 2018

If a(164) exists it's larger than 10^60. - David A. Corneth, Nov 20 2018

Links

• M. F. Hasler, Table of n, a(n) for n = 1..163 (all terms up to 10^30, and up to 10^60 according to _David A. Corneth_)
• 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 (numbers (ab+1)(ac+1), a>b>c), A320883 (subsequence of 3-smooth terms), A051037 (5-smooth numbers).

Programs

Mathematica

(* This is only a recomputation of the existing data section. *)
jmax = 12; kmax = 8; lmax = 5; max = 200000;
r[j_, k_, l_] := r[j, k, l] = If[2^j*3^k*5^l > max, Return[False], Reduce[a > b > c > 0 && (a b + 1)(a c + 1) == 2^j*3^k*5^l, {a, b, c}, Integers]];
rea = Reap[Do[rr = r[j, k, l]; If[rr =!= False, res = {j, k, l, 2^j*3^k*5^l}; Print[res]; Sow[res]], {j, 0, jmax}, {k, 0, kmax}, {l, 0, lmax}]][[2, 1]] //Union;
Print["min = ", Min /@ Transpose[rea], " max = ", Max /@ Transpose[rea]];
Sort[rea[[All, 4]]] (* Jean-François Alcover, Dec 05 2018 *)

PARI

is_A320884(n)={vecmax(factor(n,5)[,1])<6 && is_A180045(n)}
A320884=select( is_A180045, A051037_list(1e30))

Formula

Intersection of A051037 and A180045.