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%I A300835 #9 Mar 19 2018 22:15:19
%S A300835 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 A300835 21,2,22,23,24,25,26,2,27,28,29,2,30,2,31,32,33,2,34,7,35,36,37,2,38,
%U A300835 39,40,41,42,2,43,2,44,45,46,47,48,2,49,50,51,2,52,2,53,54,55,56,57,2,58,59,60,2,61,41,62,63,64,2,65,66,67,68,69
%N A300835 Restricted growth sequence transform of A300834, product_{d|n, d<n} A019565(A003714(d)); Filter sequence related to Zeckendorf-representations of proper divisors of n.
%C A300835 For all i, j: a(i) = a(j) => A001065(i) = A001065(j).
%C A300835 For all i, j: a(i) = a(j) => A300836(i) = A300836(j).
%H A300835 Antti Karttunen, <a href="/A300835/b300835.txt">Table of n, a(n) for n = 1..65537</a>
%e A300835 For cases n=10 and 49, we see that 10 has proper divisors 1, 2 and 5 and these have Zeckendorf-representations (A014417) 1, 10 and 1000, while 49 has proper divisors 1 and 7 and these have Zeckendorf-representations 1 and 1010. When these Zeckendorf-representations are summed (columnwise without carries), result in both cases is 1011, thus a(10) = a(49).
%o A300835 (PARI)
%o A300835 up_to = 65537;
%o A300835 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 A300835 write_to_bfile(start_offset,vec,bfilename) = { for(n=1, length(vec), write(bfilename, (n+start_offset)-1, " ", vec[n])); }
%o A300835 A072649(n) = { my(m); if(n<1, 0, m=0; until(fibonacci(m)>n, m++); m-2); }; \\ From A072649
%o A300835 A003714(n) = { my(s=0,w); while(n>2, w = A072649(n); s += 2^(w-1); n -= fibonacci(w+1)); (s+n); }
%o A300835 A019565(n) = {my(j,v); factorback(Mat(vector(if(n, #n=vecextract(binary(n), "-1..1")), j, [prime(j), n[j]])~))}; \\ From A019565
%o A300835 A300834(n) = { my(m=1); fordiv(n,d,if(d < n,m *= A019565(A003714(d)))); m; };
%o A300835 write_to_bfile(1,rgs_transform(vector(up_to,n,A300834(n))),"b300835.txt");
%Y A300835 Cf. A003714, A014417, A019565, A300834, A300836.
%Y A300835 Cf. also A293215, A293217, A293223, A293224, A293232, A300833 for similar filtering sequences.
%K A300835 nonn
%O A300835 1,2
%A A300835 _Antti Karttunen_, Mar 18 2018

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