login
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).
13
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, 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
OFFSET
1,2
COMMENTS
Restricted growth sequence transform of triple [A300830(n), A300831(n), A300832(n)].
For all i, j:
a(i) = a(j) => A293215(i) = A293215(j) => A001065(i) = A001065(j).
a(i) = a(j) => A051953(i) = A051953(j).
a(i) = a(j) => A295885(i) = A295885(j).
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.
LINKS
EXAMPLE
a(39) = a(55) (= 28) as A300830(39) = A300830(55) = 1, A300831(39) = A300831(55) = 2 and A300832(39) = A300832(55) = 420.
PROG
(PARI)
allocatemem(2^30);
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; };
write_to_bfile(start_offset, vec, bfilename) = { for(n=1, length(vec), write(bfilename, (n+start_offset)-1, " ", vec[n])); }
A019565(n) = {my(j, v); factorback(Mat(vector(if(n, #n=vecextract(binary(n), "-1..1")), j, [prime(j), n[j]])~))}; \\ From A019565
A300830(n) = { my(m=1); fordiv(n, d, if(!moebius(n/d), m *= A019565(d))); m; };
A300831(n) = { my(m=1); fordiv(n, d, if((d < n)&&(1==moebius(n/d)), m *= A019565(d))); m; };
A300832(n) = { my(m=1); fordiv(n, d, if(-1==moebius(n/d), m *= A019565(d))); m; };
Aux300833(n) = [A300830(n), A300831(n), A300832(n)];
write_to_bfile(1, rgs_transform(vector(up_to, n, Aux300833(n))), "b300833.txt");
CROSSREFS
KEYWORD
nonn
AUTHOR
Antti Karttunen, Mar 16 2018
STATUS
approved