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

%I A179938 #21 Nov 28 2020 03:56:45
%S A179938 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,
%T A179938 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,
%U A179938 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
%N A179938 Third largest prime factor of numbers that are divisible by at least three different primes (A000977).
%C A179938 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, ...
%e A179938 a(1) = 2 because 30 = 2 * 3 * 5 has third largest prime factor 2.
%e A179938 a(2) = 2 because 42 = 2 * 3 * 7 has third largest prime factor 2.
%e A179938 a(3) = 2 because 60 = 2 * 2 * 3 * 5 has both third and fourth largest prime factor 2.
%e A179938 a(8) = 3 because 90 = 2 * 3 * 3 * 5 has both second and third largest prime factor 3.
%p A179938 b:= proc(n) option remember; local k;
%p A179938       if n=1 then 30
%p A179938       else for k from b(n-1)+1 while
%p A179938               nops(ifactors(k)[2])<3 do od;
%p A179938            k
%p A179938       fi
%p A179938     end:
%p A179938 a:= n-> sort(map(x-> x[1]$x[2], ifactors(b(n))[2]))[-3]:
%p A179938 seq(a(n), n=1..120);
%t A179938 b[n_] := b[n] = Module[{k}, If[n==1, 30, For[k = b[n-1]+1, PrimeNu[k] < 3, k++]; k]];
%t A179938 a[n_] := (Table[#[[1]], {#[[2]]}]& /@ FactorInteger[b[n]] // Flatten // Sort)[[-3]];
%t A179938 Array[a, 120] (* _Jean-François Alcover_, Nov 28 2020, after _Alois P. Heinz_ *)
%Y A179938 Cf. A000040, A000977, A001221, A002808, A033992, A007774, A000961, A033993, A051270, A087040, A088739, A179312.
%K A179938 nonn,easy
%O A179938 1,1
%A A179938 _Jonathan Vos Post_, Jan 12 2011
%E A179938 Edited by _Alois P. Heinz_, Jan 14 2011