cp's OEIS Frontend

A320014 Filter sequence combining the binary expansions of proper divisors of n, grouped by their residue classes mod 3.

%I A320014 #22 Jan 19 2019 04:15:43
%S A320014 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 A320014 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 A320014 40,41,42,43,2,44,2,45,46,47,48,49,2,50,51,52,2,53,2,54,55,56,57,58,2,59,60,61,2,62,63,64,65,66,2,67,68,69,70,71,72,73,2,74,75
%N A320014 Filter sequence combining the binary expansions of proper divisors of n, grouped by their residue classes mod 3.
%C A320014 Restricted growth sequence transform of triple [A319990(n), A319991(n), A319992(n)], or equally, of ordered pair [A320010(n), A320013(n)].
%C A320014 Apart from trivial cases of primes, all other duplicates in range 1 .. 65537 seem to be squarefree semiprimes of the form 3k+1, i.e., both prime factors are either of the form 3k+1 or of the form 3k+2. Question: Is there any reason that more complicated cases would not occur later?
%C A320014 For all i, j: a(i) = a(j) => A293215(i) = A293215(j).
%C A320014 Differs from A319693 first for n = 108. - _Georg Fischer_, Oct 16 2018
%H A320014 Antti Karttunen, <a href="/A320014/b320014.txt">Table of n, a(n) for n = 1..65537</a>
%e A320014 The first set of numbers that forms a nontrivial equivalence class is [295, 583, 799, 943] = [5*59, 11*53, 17*47, 23*41]. The prime factors in these are all of the form 3k+2, and when the binary expansions of the factors (like e.g., "101" for 5 and "111011" for 59 or "10111" for 23 and "101001" for 41) are overlaid, the resulting bit vector is always [1, 1, 1, 1, 1, 1^2], with the least significant bit-position containing 2 copies of 1's. Thus we have a(295) = a(583) = a(799) = a(943).
%o A320014 (PARI)
%o A320014 up_to = 65537;
%o A320014 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 A320014 A019565(n) = {my(j,v); factorback(Mat(vector(if(n, #n=vecextract(binary(n), "-1..1")), j, [prime(j), n[j]])~))}; \\ From A019565
%o A320014 A319990(n) = { my(m=1); fordiv(n,d,if((d<n)&&(0==(d%3)),m *= A019565(d))); m; };
%o A320014 A319991(n) = { my(m=1); fordiv(n,d,if((d<n)&&(1==(d%3)),m *= A019565(d))); m; };
%o A320014 A319992(n) = { my(m=1); fordiv(n,d,if((d<n)&&(2==(d%3)),m *= A019565(d))); m; };
%o A320014 v320014 = rgs_transform(vector(up_to,n,[A319990(n),A319991(n),A319992(n)]));
%o A320014 A320014(n) = v320014[n];
%Y A320014 Cf. A019565, A319990, A319991, A319992, A320010, A320011, A320012, A320013.
%Y A320014 Cf. also A293214, A293215, A293226, A300833.
%Y A320014 Differs from A305800 for the first time at n=583, where a(583) = 234, while A305800(478).
%K A320014 nonn
%O A320014 1,2
%A A320014 _Antti Karttunen_, Oct 03 2018