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

Author

Antti Karttunen, Mar 16 2018

Keywords

• nonn

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.

Examples

			a(39) = a(55) (= 28) as A300830(39) = A300830(55) = 1, A300831(39) = A300831(55) = 2 and A300832(39) = A300832(55) = 420.
		

Links

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

Crossrefs

Cf. A019565, A051953, A300831, A300832.

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

Programs

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");