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A292584 Compound filter: a(n) = P(A292583(n), A292585(n)), where P(n,k) is sequence A000027 used as a pairing function.

%I A292584 #25 Sep 27 2017 09:20:11
%S A292584 1,2,5,2,8,5,13,2,4,8,19,5,25,13,9,2,32,4,41,8,12,19,51,5,16,25,5,13,
%T A292584 72,9,85,2,18,32,14,4,98,41,26,8,112,12,128,19,8,51,145,5,46,16,33,25,
%U A292584 180,5,18,13,49,72,200,9,220,85,13,2,24,18,242,32,60,14,265,4,288,98,8,41,19,26,313,8,4,112,339,12,33,128,62,19,365,8,25,51,84,145
%N A292584 Compound filter: a(n) = P(A292583(n), A292585(n)), where P(n,k) is sequence A000027 used as a pairing function.
%C A292584 From _Antti Karttunen_, Sep 25 2017: (Start)
%C A292584 Some of the sequences this filter matches to:
%C A292584 For all i, j: a(i) = a(j) => A053866(i) = A053866(j).
%C A292584 For all i, j: a(i) = a(j) => A061395(i) = A061395(j). (also A006530, etc.)
%C A292584 For all i, j: a(i) = a(j) => A292378(i) = A292378(j).
%C A292584 The latter two implications follow simply because:
%C A292584   A001222(A278222(A292383(n))) = A000120(A292383(n)) = A292377(n)
%C A292584 and, similarly, for n > 1,
%C A292584   A001222(A278222(A292385(n))) = A000120(A292381(n)) = A292375(n),
%C A292584 and the sum of A292375(n) and A292377(n) is A061395(n) [index of the largest prime dividing n], while A292378 has been defined as 1 + their difference.
%C A292584 The case A053866 follows because of the component A292583, see comments under that entry. (End)
%H A292584 Antti Karttunen, <a href="/A292584/b292584.txt">Table of n, a(n) for n = 1..16384</a>
%H A292584 <a href="/index/Pri#prime_indices">Index entries for sequences computed from indices in prime factorization</a>
%F A292584 a(n) = (1/2)*(2 + ((A292583(n)+A292585(n))^2) - A292583(n) - 3*A292585(n)).
%Y A292584 Cf. A292383, A292385, A292583, A292585.
%Y A292584 Cf. also A006530, A061395, A292378 (some of the matched sequences).
%K A292584 nonn
%O A292584 1,2
%A A292584 _Antti Karttunen_, Sep 20 2017