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

28, 126, 175, 280, 336, 378, 441, 560, 630, 672, 1225, 1470, 1680, 1701, 1792, 2016, 2520, 2835, 3136, 3969, 4200, 5250, 5600, 6860, 7840, 7875, 8400, 8960, 9072, 9408, 11025, 11340, 12096, 13125, 15120, 17640, 19845, 20160, 21000, 23520, 24696, 27440, 30625, 32928, 35000

Offset: 1

Author

M. F. Hasler, Nov 21 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 = 7 without those for p = 5 (A320884), and is therefore finite.

Links

• David A. Corneth, Table of n, a(n) for n = 1..910 (first 671 terms by Maximilian Hasler; terms < 10^35, the largest of which is ~2.6*10^31)
• 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. A080194 (greatest prime factor = 7).

Cf. A180045 (numbers (ab+1)(ac+1), a>b>c>0), A320883 (subsequence of 3-smooth terms), A320884 (subsequence of 5-smooth terms).

Programs

Mathematica

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 *)

PARI

is_A320885(n)={vecmax(factor(n,7)[,1])==7 && is_A180045(n)}
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().

Formula

Intersection of A080194 (gpf(n) = 7) and A180045 ((ab+1)(ac+1)).