A180045 Numbers of the form (ab + 1)(ac + 1) with a > b > c > 0.
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
Keywords
Examples
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
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.
Crossrefs
Programs
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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 *) -
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
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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
Comments