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A320884
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5-smooth numbers of the form (ab+1)(ac+1), a > b > c > 0.
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5
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45, 96, 120, 225, 288, 540, 640, 1080, 1200, 1920, 2160, 3888, 4000, 4500, 4608, 5760, 6480, 7200, 8640, 9600, 10935, 16875, 18225, 25000, 25600, 27000, 28800, 30720, 31104, 38400, 46080, 48600, 69984, 75000, 81000, 91125, 97200, 102400, 112500, 115200, 164025, 184320
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OFFSET
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1,1
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COMMENTS
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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 = 5, and is therefore finite.
Can someone prove that a(163) = 3327916660110655488000000000 = (16775191*16038089 + 1)(16775191*737369 + 1) = 2^42 * 3^18 * 5^9 is the last term? - M. F. Hasler, Nov 19 2018
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LINKS
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FORMULA
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MATHEMATICA
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(* This is only a recomputation of the existing data section. *)
jmax = 12; kmax = 8; lmax = 5; max = 200000;
r[j_, k_, l_] := r[j, k, l] = If[2^j*3^k*5^l > max, Return[False], Reduce[a > b > c > 0 && (a b + 1)(a c + 1) == 2^j*3^k*5^l, {a, b, c}, Integers]];
rea = Reap[Do[rr = r[j, k, l]; If[rr =!= False, res = {j, k, l, 2^j*3^k*5^l}; Print[res]; Sow[res]], {j, 0, jmax}, {k, 0, kmax}, {l, 0, lmax}]][[2, 1]] //Union;
Print["min = ", Min /@ Transpose[rea], " max = ", Max /@ Transpose[rea]];
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PROG
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CROSSREFS
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Cf. A180045 (numbers (ab+1)(ac+1), a>b>c), A320883 (subsequence of 3-smooth terms), A051037 (5-smooth numbers).
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KEYWORD
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nonn,fini
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AUTHOR
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STATUS
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approved
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