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A320014
Filter sequence combining the binary expansions of proper divisors of n, grouped by their residue classes mod 3.
5
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, 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, 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
OFFSET
1,2
COMMENTS
Restricted growth sequence transform of triple [A319990(n), A319991(n), A319992(n)], or equally, of ordered pair [A320010(n), A320013(n)].
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?
For all i, j: a(i) = a(j) => A293215(i) = A293215(j).
Differs from A319693 first for n = 108. - Georg Fischer, Oct 16 2018
LINKS
EXAMPLE
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).
PROG
(PARI)
up_to = 65537;
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; };
A019565(n) = {my(j, v); factorback(Mat(vector(if(n, #n=vecextract(binary(n), "-1..1")), j, [prime(j), n[j]])~))}; \\ From A019565
A319990(n) = { my(m=1); fordiv(n, d, if((d<n)&&(0==(d%3)), m *= A019565(d))); m; };
A319991(n) = { my(m=1); fordiv(n, d, if((d<n)&&(1==(d%3)), m *= A019565(d))); m; };
A319992(n) = { my(m=1); fordiv(n, d, if((d<n)&&(2==(d%3)), m *= A019565(d))); m; };
v320014 = rgs_transform(vector(up_to, n, [A319990(n), A319991(n), A319992(n)]));
A320014(n) = v320014[n];
CROSSREFS
Differs from A305800 for the first time at n=583, where a(583) = 234, while A305800(478).
Sequence in context: A373150 A300243 A300241 * A319693 A296073 A317943
KEYWORD
nonn
AUTHOR
Antti Karttunen, Oct 03 2018
STATUS
approved