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).

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

Antti Karttunen, Table of n, a(n) for n = 1..65537

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

Cf. A019565, A051953, A300831, A300832.

Cf. also A293214, A293215, A293226, A295885, A300825.

Sequence in context: A305895 A327931 A369050 * A300243 A300241 A320014

Adjacent sequences: A300830 A300831 A300832 * A300834 A300835 A300836

Keyword

nonn

Author

Antti Karttunen, Mar 16 2018

Status

approved