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A286358 Compound filter: a(n) = P(A286357(n), A161942(n)), where P(n,k) is sequence A000027 used as a pairing function.

%I A286358 #12 Feb 16 2025 08:33:44
%S A286358 1,4,6,22,8,13,10,106,79,47,13,39,30,19,19,466,47,742,24,233,21,58,19,
%T A286358 139,466,233,32,49,122,70,21,1954,26,380,26,4096,192,139,49,1037,233,
%U A286358 34,81,256,782,70,26,531,1597,4279,70,1227,380,157,70,157,41,1037,139,280,498,34,124,8002,256,83,174,2018,34,83,70,18916,705,1655,531,669,34,280,41
%N A286358 Compound filter: a(n) = P(A286357(n), A161942(n)), where P(n,k) is sequence A000027 used as a pairing function.
%C A286358 Partitions natural numbers to the same equivalence classes as A000203. That is, for all i, j: a(i) = a(j) <=> A000203(i) = A000203(j). This follows because both A161942(n) and A286357(n) can be (are) defined as functions of A000203, and on the other hand, A000203(n) can be uniquely reconstructed from A161942(n) and A286357(n), thus from a(n).
%H A286358 Antti Karttunen, <a href="/A286358/b286358.txt">Table of n, a(n) for n = 1..10000</a>
%H A286358 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/PairingFunction.html">Pairing Function</a>
%H A286358 <a href="/index/Si#SIGMAN">Index entries for sequences related to sigma(n)</a>
%F A286358 a(n) = (1/2)*(2 + ((A286357(n)+A161942(n))^2) - A286357(n) - 3*A161942(n)).
%o A286358 (PARI)
%o A286358 A001511(n) = (1+valuation(n,2));
%o A286358 A000265(n) = (n >> valuation(n, 2));
%o A286358 A161942(n) = A000265(sigma(n));
%o A286358 A286357(n) = A001511(sigma(n));
%o A286358 A286358(n) = (1/2)*(2 + ((A286357(n)+A161942(n))^2) - A286357(n) - 3*A161942(n));
%o A286358 for(n=1, 10000, write("b286358.txt", n, " ", A286358(n)));
%o A286358 (Scheme) (define (A286358 n) (* (/ 1 2) (+ (expt (+ (A286357 n) (A161942 n)) 2) (- (A286357 n)) (- (* 3 (A161942 n))) 2)))
%Y A286358 Cf. A000027, A000203, A161942, A286034, A286260, A286357, A286359, A286360, A286460.
%K A286358 nonn
%O A286358 1,2
%A A286358 _Antti Karttunen_, May 10 2017