A300832 -id:A300832 - cp's OEIS frontend

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

Antti Karttunen, Mar 16 2018

nonn

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.

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

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

Cf. A019565, A051953, A300831, A300832.

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

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");

A319682 Restricted growth sequence transform of A300832.

Original entry on oeis.org

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, 12, 18, 19, 2, 20, 2, 21, 22, 23, 24, 25, 2, 26, 27, 28, 2, 29, 2, 30, 31, 32, 2, 33, 7, 34, 35, 25, 2, 36, 27, 37, 38, 39, 2, 40, 2, 41, 42, 43, 44, 45, 2, 46, 47, 48, 2, 49, 2, 50, 51, 52, 53, 54, 2, 55, 56, 57, 2, 58, 59, 60, 61, 62, 2, 63, 64, 65, 66, 67, 68, 69, 2, 70, 71, 72, 2, 73, 2, 49

Offset: 1

Antti Karttunen, Sep 29 2018

nonn

For all i, j: a(i) = a(j) => A318320(i) = A318320(j).

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

Cf. A300832, A319681.

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
A300832(n) = { my(m=1); fordiv(n,d,if(-1==moebius(n/d), m *= A019565(d))); m; };
v319682 = rgs_transform(vector(up_to,n,A300832(n)));
A319682(n) = v319682[n];

A293214 a(n) = Product_{d|n, dA019565(d).

Original entry on oeis.org

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

Offset: 1

Antti Karttunen, Oct 03 2017

Antti Karttunen, Table of n, a(n) for n = 1..1024
Index entries for sequences related to binary expansion of n

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.

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; };

a(n) = Product_{d|n, dA019565(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).

A300831 a(n) = Product_{d|n, dA019565(d)^[moebius(n/d) = +1].

Original entry on oeis.org

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

Offset: 1

Antti Karttunen, Mar 16 2018

nonn

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

Cf. A008683, A008966, A019565.

Cf. also A293214, A300830, A300832, A300833.

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; };

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

A300830 a(n) = Product_{d|n} A019565(d)^(1-A008966(n/d)).

Original entry on oeis.org

1, 1, 1, 2, 1, 1, 1, 6, 2, 1, 1, 12, 1, 1, 1, 30, 1, 6, 1, 20, 1, 1, 1, 540, 2, 1, 12, 60, 1, 1, 1, 210, 1, 1, 1, 2520, 1, 1, 1, 1260, 1, 1, 1, 84, 20, 1, 1, 94500, 2, 6, 1, 140, 1, 540, 1, 18900, 1, 1, 1, 25200, 1, 1, 60, 2310, 1, 1, 1, 44, 1, 1, 1, 8731800, 1, 1, 12, 132, 1, 1, 1, 346500, 168, 1, 1, 39600, 1, 1, 1, 41580, 1, 1260

Offset: 1

Antti Karttunen, Mar 16 2018

nonn

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

Cf. A008683, A008966, A019565.

Cf. also A293214, A300831, A300832, A300833.

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; };

a(n) = Product_{d|n} A019565(d)^(1-abs(A008683(n/d))).

a(n) = A293214(n) / (A300831(n)*A300832(n)).

A329352 a(n) = Product_{d|n} A019565(d)^A010051(n/d).

Original entry on oeis.org

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, 66150, 2, 11, 252, 66, 300, 1155, 2, 198, 420, 385, 2, 173250, 2, 825, 2940, 990, 2, 847, 30, 3234, 132, 1155, 2, 15246, 420, 2695, 396, 2310, 2, 2223375, 2, 6930, 1540, 13, 700, 64350, 2, 195, 1980, 171990, 2, 5005, 2, 390, 32340, 975, 1260

Offset: 1

Antti Karttunen, Nov 12 2019

nonn

			The divisors of 30 are [1, 2, 3, 5, 6, 10, 15, 30], of which only d = 6, 10 and 15 are such that 30/d is a prime, thus a(n) = A019565(6) * A019565(10) * A019565(15) = 15 * 21 * 210 = 66150.

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

Cf. A010051, A019565, A048675, A069359, A329353 (rgs-transform).
Cf. also A329350.
Differs from A300832 for the first time at n=30, where a(30) = 66150, while A300832(30) = 132300.

A019565(n) = {my(j,v); factorback(Mat(vector(if(n, #n=vecextract(binary(n), "-1..1")), j, [prime(j), n[j]])~))}; \\ From A019565
A329352(n) = { my(m=1); fordiv(n,d,if(isprime(n/d), m *= A019565(d))); (m); };

a(n) = Product_{d|n} A019565(d)^A010051(n/d).

For all n, A048675(a(n)) = A069359(n).

A320017 a(1) = 1; for n > 1, a(n) = Product_{d|n} A019565(d)^[moebius(d) = +1].

Original entry on oeis.org

1, 2, 2, 2, 2, 30, 2, 2, 2, 42, 2, 30, 2, 210, 420, 2, 2, 30, 2, 42, 220, 330, 2, 30, 2, 462, 2, 210, 2, 132300, 2, 2, 52, 78, 156, 30, 2, 390, 780, 42, 2, 346500, 2, 330, 420, 2730, 2, 30, 2, 42, 1716, 462, 2, 30, 8580, 210, 4004, 6006, 2, 132300, 2, 30030, 220, 2, 68, 128700, 2, 78, 340, 343980, 2, 30, 2, 714, 420, 390, 2380, 2702700, 2, 42, 2

Offset: 1

Antti Karttunen, Nov 08 2018

nonn

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

Cf. A019565, A320018 (rgs-transform).

Cf. also A300831, A300832.

A019565(n) = {my(j,v); factorback(Mat(vector(if(n, #n=vecextract(binary(n), "-1..1")), j, [prime(j), n[j]])~))}; \\ From A019565
A320017(n) = if(1==n,n,my(m=1); fordiv(n,d,if(1==moebius(d), m *= A019565(d))); (m));

a(1) = 1; for n > 1, a(n) = Product_{d|n} A019565(d)^[A008683(d) > 0].

For n >= 2, A048675(a(n)) = A318674(n).

cp's OEIS Frontend

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).

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A319682 Restricted growth sequence transform of A300832.

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A293214 a(n) = Product_{d|n, dA019565(d).

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A300831 a(n) = Product_{d|n, dA019565(d)^[moebius(n/d) = +1].

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A300830 a(n) = Product_{d|n} A019565(d)^(1-A008966(n/d)).

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A329352 a(n) = Product_{d|n} A019565(d)^A010051(n/d).

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A320017 a(1) = 1; for n > 1, a(n) = Product_{d|n} A019565(d)^[moebius(d) = +1].

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