A064708 Initial term of run of (at least) n consecutive numbers with just 2 distinct prime factors.

6, 14, 20, 33, 54, 91, 141, 141

Offset: 1

Author

Robert G. Wilson v, Oct 13 2001

Keywords

• nonn
• fini

Comments

It can be shown by an application of Mihailescu's theorem that a(12) does not exist, since then there would be two 3-smooth numbers close together (it suffices to check up to 2*3^3).

If a(9) exists, it is greater than 10^30. - Don Reble, Mar 02 2003

If a(9) exists, it is greater than 10^3000. - Charles R Greathouse IV, Apr 22 2009

Eggleton and MacDougall prove that no terms exist beyond a(9) and conjecture that a(9) does not exist. - Jason Kimberley, Jul 08 2017

Examples

			6 = 2*3; 14 = 2*7 and 15 = 3*5; 20 = 2^2*5, 21 = 3*7 and 22 = 2*11; 33 = 3*11, 34 = 2*17, 35 = 5*7 and 36 = (2*3)^2; etc.
		

Links

• Roger B. Eggleton and James A. MacDougall, Consecutive integers with equally many principal divisors, Math. Mag. 81 (2008), 235-248. [_T. D. Noe_, Oct 13 2008]

Crossrefs

Cf. A064709.

Programs

Mathematica

Function[s, Function[t, Reverse@ FoldList[If[#2 > #1, #1, #2] &, Reverse[#]] &@ Map[t[[First@ FirstPosition[t[[All, -1]], k_ /; k == #] ]] &, Range[0, Max@ t[[All, -1]] ] ][[All, 1]] ]@ Join[{{First@ s, 0}, {#[[1, 1, 1]], 1}}, Rest@ Map[{#[[1, 1]], Length@ # + 1} &, #, {1}]] &@ SplitBy[Partition[Select[#, Last@ # == 1 &][[All, 1]], 2, 1], Differences] &@ Map[{First@ #, First@ Differences@ #} &, Partition[s, 2, 1]]]@ Select[Range[10^5], PrimeNu[#] == 2 &] (* Michael De Vlieger, Jul 17 2017 *)