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Search: a300832 -id:a300832
Displaying 1-7 of 7 results found.
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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
A319682
Restricted growth sequence transform of A300832.
+20
4
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
(list; graph; refs; listen; history; text; internal format)
OFFSET
1,2
COMMENTS
For all i, j: a(i) = a(j) => A318320(i) = A318320(j).
LINKS
Antti Karttunen, Table of n, a(n) for n = 1..65537
PROG
(PARI)
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];
CROSSREFS
Cf. A300832, A319681.
KEYWORD
nonn
AUTHOR
Antti Karttunen, Sep 29 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
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
A300830
a(n) = Product_{d|n} A019565(d)^(1-A008966(n/d)).
+10
5
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
(list; graph; refs; listen; history; text; internal format)
OFFSET
1,4
LINKS
Antti Karttunen, Table of n, a(n) for n = 1..8192
FORMULA
a(n) = Product_{d|n} A019565(d)^(1-abs(A008683(n/d))).
a(n) = A293214(n) / (A300831(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
A300830(n) = { my(m=1); fordiv(n, d, if(!moebius(n/d), m *= A019565(d))); m; };
CROSSREFS
Cf. A008683, A008966, A019565.
Cf. also A293214, A300831, A300832, A300833.
KEYWORD
nonn
AUTHOR
Antti Karttunen, Mar 16 2018
STATUS
approved
A329352
a(n) = Product_{d|n} A019565(d)^A010051(n/d).
+10
5
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
(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) = Product_{d|n} A019565(d)^A010051(n/d).
For all n, A048675(a(n)) = A069359(n).
EXAMPLE
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.
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
A329352(n) = { my(m=1); fordiv(n, d, if(isprime(n/d), m *= A019565(d))); (m); };
CROSSREFS
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.
KEYWORD
nonn
AUTHOR
Antti Karttunen, Nov 12 2019
STATUS
approved
A320017
a(1) = 1; for n > 1, a(n) = Product_{d|n} A019565(d)^[moebius(d) = +1].
+10
3
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
(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(1) = 1; for n > 1, a(n) = Product_{d|n} A019565(d)^[A008683(d) > 0].
For n >= 2, A048675(a(n)) = A318674(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
A320017(n) = if(1==n, n, my(m=1); fordiv(n, d, if(1==moebius(d), m *= A019565(d))); (m));
CROSSREFS
Cf. A019565, A320018 (rgs-transform).
Cf. also A300831, A300832.
KEYWORD
nonn
AUTHOR
Antti Karttunen, Nov 08 2018
STATUS
approved
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