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A180045
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Numbers of the form (ab + 1)(ac + 1) with a > b > c > 0.
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9
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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
(list;
graph;
refs;
listen;
history;
text;
internal format)
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OFFSET
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1,1
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COMMENTS
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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
The number of p-smooth terms appears to be (0, 12, 163, ...) for p = prime(1, 2, 3, ...). - M. F. Hasler, Nov 20 2018
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LINKS
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EXAMPLE
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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
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MATHEMATICA
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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 *)
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PROG
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(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
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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