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A300833 Filter sequence combining A300830(n), A300831(n) and A300832(n), three products formed from such proper divisors d of n for which mu(n/d) = 0, +1 or -1 respectively, where mu is Möbius mu function (A008683).

%I A300833 #25 Jan 19 2019 04:15:43
%S A300833 1,2,2,3,2,4,2,5,6,7,2,8,2,9,10,11,2,12,2,13,14,15,2,16,17,18,19,20,2,
%T A300833 21,2,22,23,24,25,26,2,27,28,29,2,30,2,31,32,33,2,34,35,36,37,38,2,39,
%U A300833 28,40,41,42,2,43,2,44,45,46,47,48,2,49,50,51,2,52,2,53,54,55,56,57,2,58,59,60,2,61,62,63,64,65,2,66,67,68,69,70
%N A300833 Filter sequence combining A300830(n), A300831(n) and A300832(n), three products formed from such proper divisors d of n for which mu(n/d) = 0, +1 or -1 respectively, where mu is Möbius mu function (A008683).
%C A300833 Restricted growth sequence transform of triple [A300830(n), A300831(n), A300832(n)].
%C A300833 For all i, j:
%C A300833   a(i) = a(j) => A293215(i) = A293215(j) => A001065(i) = A001065(j).
%C A300833   a(i) = a(j) => A051953(i) = A051953(j).
%C A300833   a(i) = a(j) => A295885(i) = A295885(j).
%C A300833 Apparently this is also the restricted growth sequence transform of ordered pair [A300831(n), A300832(n)], which is true if it holds that whenever we have A300831(i) = A300831(j) and A300832(i) = A300832(j) for any i, j, then also A300830(i) = A300830(j). This has been checked for the first 65537 terms.
%H A300833 Antti Karttunen, <a href="/A300833/b300833.txt">Table of n, a(n) for n = 1..65537</a>
%e A300833 a(39) = a(55) (= 28) as A300830(39) = A300830(55) = 1, A300831(39) = A300831(55) = 2 and A300832(39) = A300832(55) = 420.
%o A300833 (PARI)
%o A300833 allocatemem(2^30);
%o A300833 up_to = 65537;
%o A300833 rgs_transform(invec) = { my(om = Map(), outvec = vector(length(invec)), u=1); for(i=1, length(invec), if(mapisdefined(om,invec[i]), my(pp = mapget(om, invec[i])); outvec[i] = outvec[pp] , mapput(om,invec[i],i); outvec[i] = u; u++ )); outvec; };
%o A300833 write_to_bfile(start_offset,vec,bfilename) = { for(n=1, length(vec), write(bfilename, (n+start_offset)-1, " ", vec[n])); }
%o A300833 A019565(n) = {my(j,v); factorback(Mat(vector(if(n, #n=vecextract(binary(n), "-1..1")), j, [prime(j), n[j]])~))}; \\ From A019565
%o A300833 A300830(n) = { my(m=1); fordiv(n,d,if(!moebius(n/d),m *= A019565(d))); m; };
%o A300833 A300831(n) = { my(m=1); fordiv(n,d,if((d < n)&&(1==moebius(n/d)), m *= A019565(d))); m; };
%o A300833 A300832(n) = { my(m=1); fordiv(n,d,if(-1==moebius(n/d), m *= A019565(d))); m; };
%o A300833 Aux300833(n) = [A300830(n), A300831(n), A300832(n)];
%o A300833 write_to_bfile(1,rgs_transform(vector(up_to,n,Aux300833(n))),"b300833.txt");
%Y A300833 Cf. A019565, A051953, A300831, A300832.
%Y A300833 Cf. also A293214, A293215, A293226, A295885, A300825.
%K A300833 nonn
%O A300833 1,2
%A A300833 _Antti Karttunen_, Mar 16 2018