A186927 Lesser of two consecutive 3-smooth numbers having no common divisors.

1, 2, 3, 8, 27, 243, 2048, 524288, 129140163, 68630377364883, 36472996377170786403, 19342813113834066795298816, 706965049015104706497203195837614914543357369, 13703277223523221219433362313025801636536040755174924956117940937101787

Offset: 1

Author

Charles R Greathouse IV and Reinhard Zumkeller, Mar 01 2011

Keywords

• nonn

Comments

a(n) = A003586(A186771(n)); A186928(n) = A003586(A186771(n) + 1).

Subsequence of A006899: all terms are either powers of 2 or of 3.

Najman improves an algorithm of Bauer & Bennett for computing the function that measures the minimal gap size f(k) in the sequence of integers at least one of whose prime factors exceeds k. This allows us to compute values of f(k) for larger k and obtain new values of f(k). - Jonathan Vos Post, Aug 18 2011

Links

• Charles R Greathouse IV, Table of n, a(n) for n = 1..21
• M. Bauer and M. A. Bennett, Prime factors of consecutive integers, Mathematics of Computation 77 (2008), pp. 2455-2459.
• Charles R Greathouse IV, Illustration of n, a(n) for n = 1..33
• Filip Najman, Large strings of consecutive smooth integers, Aug 18, 2011

Crossrefs

Cf. A186711.

Programs

Mathematica

smoothNumbers[p_, max_] := Module[{a, aa, k, pp, iter}, k = PrimePi[p]; aa = Array[a, k]; pp = Prime[Range[k]]; iter = Table[{a[j], 0, PowerExpand @ Log[pp[[j]], max/Times @@ (Take[pp, j - 1]^Take[aa, j - 1])]}, {j, 1, k}]; Table[Times @@ (pp^aa), Sequence @@ iter // Evaluate] // Flatten // Sort]; sn = smoothNumbers[3, 10^100]; Reap[For[i = 1, i <= Length[sn] - 1, i++, If[CoprimeQ[sn[[i]], sn[[i + 1]]], Sow[sn[[i]]]]]][[2, 1]] (* Jean-François Alcover, Nov 11 2016 *)