|
%I #75 Dec 05 2018 08:33:47
%S 28,45,65,66,91,96,117,120,126,133,153,175,176,190,217,225,231,232,
%T 247,276,280,288,297,325,330,336,341,344,369,370,378,403,408,425,435,
%U 441,451,460,475,481,496,513,532,540,550,560,561,589,630,637,638,640,645,651,671,672,697,703,730,736,742,775,780,781,782,792,793,804,825,833,855,861,874,891,924,925,936,946,949,969,976,1001
%N Numbers of the form (ab + 1)(ac + 1) with a > b > c > 0.
%C Corvaja & Zannier show that the greatest prime factor of members of this sequence tends to infinity. In other words, for any set S of primes, only finitely many members of this sequence are S-smooth (having all their prime divisors in S).
%C 440301256704 = (2359*889 + 1)(2359*89 + 1) = 2^26 * 3^8 is in the sequence; are there any larger 3-smooth terms?
%C Similarly, 3327916660110655488000000000 = (16775191*16038089 + 1)(16775191*737369 + 1) = 2^42 * 3^18 * 5^9 is in the sequence; are there any larger 5-smooth terms? - _Charles R Greathouse IV_, Nov 02 2018
%C See A320883 for the 3-smooth and A320884 for the 5-smooth terms. - _M. F. Hasler_, Nov 19 2018
%C The number of p-smooth terms appears to be (0, 12, 163, ...) for p = prime(1, 2, 3, ...). - _M. F. Hasler_, Nov 20 2018
%H Charles R Greathouse IV, <a href="/A180045/b180045.txt">Table of n, a(n) for n = 1..10000</a>
%H P. Corvaja and U. Zannier, <a href="https://arxiv.org/abs/math/0205136">On the greatest prime factor of (ab+1)(ac+1)</a>, arXiv:math/0205136 [math.NT], 2002.
%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.
%e 1001 is a term. Checking divisors (k, m = 1001/k), where m > k, we look at the GCD = a of k=1 and m-1. For (k, m) = (11, 91), we find a = gcd(k-1 = 10 = a*c, m-1 = 90 = a*b) = 10 and the corresponding c = 1 and b = 9 meet the required a > b > c > 0. Therefore 1001 is a term. - _David A. Corneth_, Nov 21 2018
%t max = 1001; amax = Ceiling[(Sqrt[8 max + 1] - 3)/4];
%t Reap[Do[If[a > b > c > 0, m = (a b + 1)(a c + 1); If[m <= max, Sow[m]]], {a, 1, amax}, {b, 1, a-1}, {c, 1, b-1}]][[2, 1]] // Union (* _Jean-François Alcover_, Dec 05 2018 *)
%o (PARI) list(lim)=my(v=List(),t); for(c=1,sqrtnint(lim\=1,4), for(b=c+1,sqrtnint(lim\c,3), for(a=b+1,lim\(b+c), t=(a*b+1)*(a*c+1); if(t>lim,break); listput(v,t)))); Set(v); \\ edited by _Charles R Greathouse IV_, Oct 28 2018
%o (PARI) is_A180045(n)={fordiv(n, d, if(d^2>=n, return(0), d^3 > n && gcd(d-1, n\d-1)^2*d >= n, return(1)))} \\ This defines the is_A180045() function used in several other sequences. To compute a list of initial terms, use the list() function above. - _David A. Corneth_ and _M. F. Hasler_, Nov 21 2018, based on earlier code from _Charles R Greathouse IV_
%Y Cf. A320883, A320884, A320885 (subsequences of 3-smooth, 5-smooth terms and 7-smooth terms).
%K nonn
%O 1,1
%A _Charles R Greathouse IV_, Jul 06 2011
|
