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

1,1

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

Links

Table of n, a(n) for n=1..8.

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]

Example

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.

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 *)

Crossrefs

Cf. A064709.

Sequence in context: A101567 A123267 A236928 * A064709 A371396 A118129

Adjacent sequences: A064705 A064706 A064707 * A064709 A064710 A064711

Keyword

nonn,fini

Author

Robert G. Wilson v, Oct 13 2001

Status

approved