A300830 -id:A300830

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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).

+20
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

(list; graph; refs; listen; history; text; internal format)

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.

KEYWORD

nonn

AUTHOR

Antti Karttunen, Mar 16 2018

STATUS

approved

A293214

a(n) = Product_{d|n, d<n} A019565(d).

+10
25

1, 2, 2, 6, 2, 36, 2, 30, 12, 60, 2, 2700, 2, 180, 120, 210, 2, 7560, 2, 6300, 360, 252, 2, 661500, 20, 420, 168, 94500, 2, 23814000, 2, 2310, 504, 132, 600, 43659000, 2, 396, 840, 2425500, 2, 187110000, 2, 207900, 352800, 1980, 2, 560290500, 60, 194040, 264, 485100, 2, 115259760, 840, 254677500, 792, 4620, 2, 264737261250000, 2, 13860

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,2

LINKS

Antti Karttunen, Table of n, a(n) for n = 1..1024

Index entries for sequences related to binary expansion of n

FORMULA

a(n) = Product_{d|n, d<n} A019565(d).

a(n) = A300830(n) * A300831(n) * A300832(n). - Antti Karttunen, Mar 16 2018

Other identities.

For n >= 0, a(2^n) = A002110(n).

For n >= 1:

A048675(a(n)) = A001065(n).

A001222(a(n)) = A292257(n).

A007814(a(n)) = A091954(n).

A087207(a(n)) = A218403(n).

A248663(a(n)) = A227320(n).

PROG

(PARI)

A019565(n) = {my(j, v); factorback(Mat(vector(if(n, #n=vecextract(binary(n), "-1..1")), j, [prime(j), n[j]])~))}; \\ From A019565

A293214(n) = { my(m=1); fordiv(n, d, if(d < n, m *= A019565(d))); m; };

CROSSREFS

Cf. A001065, A002110, A019565, A048675, A091954, A292257, A293215 (restricted growth sequence transform).

Cf. also A293216, A293221, A293222, A293225, A293231, A293442, A300830, A300831, A300832, A300833, A300834.

KEYWORD

nonn,base

AUTHOR

Antti Karttunen, Oct 03 2017

STATUS

approved

A300832

a(n) = Product_{d|n} A019565(d)^[moebius(n/d) = -1].

+10
8

1, 2, 2, 3, 2, 18, 2, 5, 6, 30, 2, 75, 2, 90, 60, 7, 2, 210, 2, 105, 180, 126, 2, 245, 10, 210, 14, 525, 2, 132300, 2, 11, 252, 66, 300, 1155, 2, 198, 420, 385, 2, 346500, 2, 825, 2940, 990, 2, 847, 30, 3234, 132, 1155, 2, 15246, 420, 2695, 396, 2310, 2, 6670125, 2, 6930, 1540, 13, 700, 128700, 2, 195, 1980, 343980, 2, 5005, 2, 390

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,2

LINKS

Antti Karttunen, Table of n, a(n) for n = 1..8192

FORMULA

a(n) = A293214(n) / (A300830(n)*A300831(n)).

PROG

(PARI)

A019565(n) = {my(j, v); factorback(Mat(vector(if(n, #n=vecextract(binary(n), "-1..1")), j, [prime(j), n[j]])~))}; \\ From A019565

A300832(n) = { my(m=1); fordiv(n, d, if(-1==moebius(n/d), m *= A019565(d))); m; };

CROSSREFS

Cf. A008683, A008966, A019565, A293214, A300830, A300831, A300833.

KEYWORD

nonn

AUTHOR

Antti Karttunen, Mar 16 2018

STATUS

approved

A300831

a(n) = Product_{d|n, d<n} A019565(d)^[moebius(n/d) = +1].

+10
7

1, 1, 1, 1, 1, 2, 1, 1, 1, 2, 1, 3, 1, 2, 2, 1, 1, 6, 1, 3, 2, 2, 1, 5, 1, 2, 1, 3, 1, 180, 1, 1, 2, 2, 2, 15, 1, 2, 2, 5, 1, 540, 1, 3, 6, 2, 1, 7, 1, 10, 2, 3, 1, 14, 2, 5, 2, 2, 1, 1575, 1, 2, 6, 1, 2, 756, 1, 3, 2, 900, 1, 35, 1, 2, 10, 3, 2, 1260, 1, 7, 1, 2, 1, 7875, 2, 2, 2, 5, 1, 44100, 2, 3, 2, 2, 2, 11, 1, 30, 6, 21, 1, 396, 1, 5, 1800

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,6

LINKS

Antti Karttunen, Table of n, a(n) for n = 1..8192

FORMULA

a(n) = A293214(n) / (A300830(n)*A300832(n)).

PROG

(PARI)

A019565(n) = {my(j, v); factorback(Mat(vector(if(n, #n=vecextract(binary(n), "-1..1")), j, [prime(j), n[j]])~))}; \\ From A019565

A300831(n) = { my(m=1); fordiv(n, d, if((d < n)&&(1==moebius(n/d)), m *= A019565(d))); m; };

CROSSREFS

Cf. A008683, A008966, A019565.

Cf. also A293214, A300830, A300832, A300833.

KEYWORD

nonn

AUTHOR

Antti Karttunen, Mar 16 2018

STATUS

approved

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