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A180045
Numbers of the form (ab + 1)(ac + 1) with a > b > c > 0.
9
28, 45, 65, 66, 91, 96, 117, 120, 126, 133, 153, 175, 176, 190, 217, 225, 231, 232, 247, 276, 280, 288, 297, 325, 330, 336, 341, 344, 369, 370, 378, 403, 408, 425, 435, 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
OFFSET
1,1
COMMENTS
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).
440301256704 = (2359*889 + 1)(2359*89 + 1) = 2^26 * 3^8 is in the sequence; are there any larger 3-smooth terms?
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
See A320883 for the 3-smooth and A320884 for the 5-smooth terms. - M. F. Hasler, Nov 19 2018
The number of p-smooth terms appears to be (0, 12, 163, ...) for p = prime(1, 2, 3, ...). - M. F. Hasler, Nov 20 2018
LINKS
Charles R Greathouse IV, Table of n, a(n) for n = 1..10000
P. Corvaja and U. Zannier, On the greatest prime factor of (ab+1)(ac+1), arXiv:math/0205136 [math.NT], 2002.
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.
EXAMPLE
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
MATHEMATICA
max = 1001; amax = Ceiling[(Sqrt[8 max + 1] - 3)/4];
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 *)
PROG
(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
(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
CROSSREFS
Cf. A320883, A320884, A320885 (subsequences of 3-smooth, 5-smooth terms and 7-smooth terms).
Sequence in context: A349495 A169962 A219685 * A144581 A075875 A332764
KEYWORD
nonn
AUTHOR
STATUS
approved