A179938 Third largest prime factor of numbers that are divisible by at least three different primes (A000977).

2, 2, 2, 2, 2, 2, 2, 3, 2, 3, 2, 2, 2, 3, 2, 2, 2, 2, 3, 2, 2, 3, 2, 2, 2, 3, 2, 2, 2, 3, 3, 2, 3, 2, 2, 2, 2, 3, 3, 2, 2, 2, 3, 3, 2, 2, 2, 2, 3, 3, 2, 2, 2, 3, 2, 2, 3, 3, 3, 2, 2, 2, 3, 2, 2, 3, 2, 2, 3, 3, 2, 5, 2, 3, 3, 2, 2, 2, 2, 2, 3, 2, 5, 3, 3, 3, 2, 2, 2, 2, 3, 2, 3, 2, 3, 2, 2, 3, 2, 2, 2, 2, 3, 5, 2, 2, 3, 3, 3, 2, 2, 2, 2, 3, 5, 2, 2, 3, 2, 3

Offset: 1

Author

Jonathan Vos Post, Jan 12 2011

Keywords

• nonn
• easy

Comments

Third largest prime factor of numbers k such that omega(k) = A001221(k) > 2. The 3rd largest prime factor may equal the second largest. This is not identical to third largest distinct prime factor of numbers that are divisible by at least three different primes. Indices n where a(n) equals 2, 3, 5, 7, 11, 13, 17, 19, 23, ... for the first time are 1, 8, 72, 299, 905, 1718, 3302, 6020, 10330, ... the corresponding numbers from A000977 are 30, 90, 350, 1001, 2431, 4199, 7429, 12673, 20677, ...

Examples

			a(1) = 2 because 30 = 2 * 3 * 5 has third largest prime factor 2.
a(2) = 2 because 42 = 2 * 3 * 7 has third largest prime factor 2.
a(3) = 2 because 60 = 2 * 2 * 3 * 5 has both third and fourth largest prime factor 2.
a(8) = 3 because 90 = 2 * 3 * 3 * 5 has both second and third largest prime factor 3.
		

Crossrefs

Cf. A000040, A000977, A001221, A002808, A033992, A007774, A000961, A033993, A051270, A087040, A088739, A179312.

Programs

Maple

b:= proc(n) option remember; local k;
      if n=1 then 30
      else for k from b(n-1)+1 while
              nops(ifactors(k)[2])<3 do od;
           k
      fi
    end:
a:= n-> sort(map(x-> x[1]$x[2], ifactors(b(n))[2]))[-3]:
seq(a(n), n=1..120);

Mathematica

b[n_] := b[n] = Module[{k}, If[n==1, 30, For[k = b[n-1]+1, PrimeNu[k] < 3, k++]; k]];
a[n_] := (Table[#[[1]], {#[[2]]}]& /@ FactorInteger[b[n]] // Flatten // Sort)[[-3]];
Array[a, 120] (* Jean-François Alcover, Nov 28 2020, after Alois P. Heinz *)

Extensions

Edited by Alois P. Heinz, Jan 14 2011