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A292586 a(n) = A002110(A001221(n)) = product of first omega(n) primes.

%I A292586 #22 Sep 27 2017 09:20:22
%S A292586 1,2,2,2,2,6,2,2,2,6,2,6,2,6,6,2,2,6,2,6,6,6,2,6,2,6,2,6,2,30,2,2,6,6,
%T A292586 6,6,2,6,6,6,2,30,2,6,6,6,2,6,2,6,6,6,2,6,6,6,6,6,2,30,2,6,6,2,6,30,2,
%U A292586 6,6,30,2,6,2,6,6,6,6,30,2,6,2,6,2,30,6,6,6,6,2,30,6,6,6,6,6,6,2,6,6,6,2,30,2,6,30
%N A292586 a(n) = A002110(A001221(n)) = product of first omega(n) primes.
%C A292586 The connection with binary tree A005940 is explained by the fact that on a trajectory from its root (1) to any number n, the numbers of the form 4k+2 will never occur consecutively (they are only born as right children of odd numbers, while all their right descendants from then onward are multiples of four). Thus all the runs are separate runs of length one, from which follows that A278222 when applied to A292382 yields only primorials. Moreover, the steps producing 4k+2 numbers are also only steps in A005940 that add new distinct prime factors to the generated number. Thus the total number of such steps is equal to the number of distinct prime factors of the eventual n. Hence A278222(A292382(n)) = A002110(A001221(n)).
%H A292586 Antti Karttunen, <a href="/A292586/b292586.txt">Table of n, a(n) for n = 1..16384</a>
%H A292586 <a href="/index/Pri#primorial_numbers">Index entries for sequences related to primorial numbers</a>
%F A292586 a(n) = A002110(A001221(n)).
%F A292586 a(n) = A278222(A292382(n)).
%F A292586 For all n >= 1:
%F A292586 A001221(n) = A001221(a(n)) = A001222(a(n)) = A000120(A292382(n)).
%t A292586 Array[Product[Prime@ i, {i, PrimeNu@ #}] &, 105] (* _Michael De Vlieger_, Sep 25 2017 *)
%o A292586 (PARI) A292586(n) = prod(i=1, omega(n), prime(i));
%o A292586 (Scheme)
%o A292586 (define (A292586 n) (A002110 (A001221 n)))
%o A292586 (define (A292586 n) (A278222 (A292382 n)))
%Y A292586 Cf. A005940, A086568, A278222, A292382.
%Y A292586 Cf. A083399 (restricted growth transform of this sequence).
%K A292586 nonn
%O A292586 1,2
%A A292586 _Antti Karttunen_, Sep 25 2017