A320884 -id:A320884 - cp's OEIS frontend

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

Charles R Greathouse IV, Jul 06 2011

nonn

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

			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

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.

Cf. A320883, A320884, A320885 (subsequences of 3-smooth, 5-smooth terms and 7-smooth terms).

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 *)

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

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

A320883 3-smooth numbers of the form (ab+1)(ac+1), a > b > c > 0.

Original entry on oeis.org

96, 288, 3888, 4608, 31104, 69984, 2654208, 2985984, 4478976, 1088391168, 1528823808, 440301256704

Offset: 1

M. F. Hasler, Nov 19 2018

Subsequence of A320884 = 5-smooth terms of A180045, finite according to Corvaja & Zannier.

Can someone prove that a(12) = 440301256704 = (2359*889 + 1)(2359*89 + 1) = 2^26 * 3^8 is the last term?

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.

Cf. A180045 = {(ab+1)(ac+1); a > b > c > 0}, A320884 (5-smooth terms of A180045), A003586 (3-smooth numbers).

(* This is only a recomputation of the existing sequence. *)
(* Max exponents: *) jmax = 26; kmax = 12;
r[j_, k_] := Reduce[a > b > c > 0 && (a b + 1)(a c + 1) == 2^j*3^k , {a, b, c}, Integers];
Reap[Do[rr = r[j, k]; If[rr =!= False, Print[{j, k, 2^j*3^k}]; Sow[2^j*3^k]], {j, 1, jmax}, {k, 1, kmax}]][[2, 1]] // Union (* Jean-François Alcover, Dec 05 2018 *)

A320883(LIM=35,S=[])={for(m=1,LIM, for(k=0,m, is_A180045(3^k<<(m-k))&& S=setunion(S,[3^k<<(m-k)])));S} \\ Gives all terms up to 2^LIM and possibly some larger terms up to 3^LIM.
is_A320883(n)={vecmax(factor(n,3)[,1])<4 && is_A180045(n)}

Intersection of A180045 = {(ab+1)(ac+1); a > b > c > 0} and A003586 (3-smooth numbers).

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

Original entry on oeis.org

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

M. F. Hasler, Nov 21 2018

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.

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.

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).

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 *)

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().

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

A332764 7-smooth numbers of the form (ab+1)*(ac+1), a > b > c > 0.

Original entry on oeis.org

28, 45, 96, 120, 126, 175, 225, 280, 288, 336, 378, 441, 540, 560, 630, 640, 672, 1080, 1200, 1225, 1470, 1680, 1701, 1792, 1920, 2016, 2160, 2520, 2835, 3136, 3888, 3969, 4000, 4200, 4500, 4608, 5250, 5600, 5760, 6480, 6860, 7200, 7840, 7875, 8400, 8640, 8960

Offset: 1

Jon E. Schoenfield, Feb 22 2020

Sequence is finite (see comments at A320884).

David A. Corneth, Table of n, a(n) for n = 1..1072

Cf. A180045 (numbers (ab+1)(ac+1), a > b > c > 0), A320883 (subsequence of 3-smooth terms), A320884 (subsequence of 5-smooth terms), A002473 (7-smooth numbers).

Block[{nn = 9000, nm, m}, nm = Ceiling[(Sqrt[8 nn + 1] - 3)/4]; Union@ Reap[Do[If[a > b > c > 0, Set[m, (a b + 1) (a c + 1)]; If[And[m <= nn, FactorInteger[m][[-1, 1]] <= 7 ], Sow[m]]], {a, nm}, {b, a - 1}, {c, b - 1}]][[2, 1]]] (* Michael De Vlieger, Feb 25 2020, after Jean-François Alcover at A180045 *)

Intersection of A002473 and A180045.

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A180045 Numbers of the form (ab + 1)(ac + 1) with a > b > c > 0.

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A320883 3-smooth numbers of the form (ab+1)(ac+1), a > b > c > 0.

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

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A332764 7-smooth numbers of the form (ab+1)*(ac+1), a > b > c > 0.

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