A293214 -id:A293214 - cp's OEIS frontend

A300832 a(n) = Product_{d|n} A019565(d)^[moebius(n/d) = -1].

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

Offset: 1

Antti Karttunen, Mar 16 2018

nonn

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

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

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; };

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

A319991 a(n) = Product_{d|n, dA019565(d)^[1 == d mod 3].

Original entry on oeis.org

1, 2, 2, 2, 2, 2, 2, 10, 2, 2, 2, 10, 2, 60, 2, 10, 2, 2, 2, 210, 60, 2, 2, 10, 2, 140, 2, 300, 2, 42, 2, 110, 2, 2, 60, 10, 2, 132, 140, 210, 2, 60, 2, 1650, 2, 2, 2, 110, 60, 6468, 2, 700, 2, 2, 2, 115500, 132, 2, 2, 210, 2, 4620, 60, 110, 140, 330, 2, 390, 2, 1260, 2, 10, 2, 260, 308, 660, 60, 140, 2, 210210, 2, 2, 2, 115500, 2, 1092, 2

Offset: 1

Antti Karttunen, Oct 03 2018

nonn

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

Cf. A019565, A293214, A293895, A293897, A319990, A319992, A320001, A320011 (rgs-transform).

Cf. also A293221.

A019565(n) = {my(j,v); factorback(Mat(vector(if(n, #n=vecextract(binary(n), "-1..1")), j, [prime(j), n[j]])~))}; \\ This function from M. F. Hasler
A319991(n) = { my(m=1); fordiv(n,d,if((dA019565(d))); m; };

a(n) = Product_{d|n, dA019565(d)^[1 == d mod 3].
a(n) = A293214(n) / (A319990(n)*A319992(n)).
For all n >= 1:
A007814(a(n)) = A320001(n).
A048675(a(n)) = A293897(n).
A195017(a(n)) = A293895(n) mod 3.

A319992 a(n) = Product_{d|n, dA019565(d)^[2 == d mod 3].

Original entry on oeis.org

1, 1, 1, 3, 1, 3, 1, 3, 1, 30, 1, 3, 1, 3, 10, 21, 1, 3, 1, 30, 1, 126, 1, 21, 10, 3, 1, 315, 1, 30, 1, 21, 42, 66, 10, 3, 1, 3, 1, 11550, 1, 315, 1, 126, 10, 990, 1, 21, 1, 30, 22, 693, 1, 3, 420, 2205, 1, 2310, 1, 1650, 1, 3, 1, 273, 10, 126, 1, 66, 330, 245700, 1, 21, 1, 3, 10, 585, 42, 693, 1, 11550, 1, 546, 1, 315, 220, 3, 770

Offset: 1

Antti Karttunen, Oct 03 2018

nonn

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

Cf. A019565, A293214, A293896, A293898, A319990, A319991, A320005, A320012 (rgs-transform).

Cf. also A293222.

A019565(n) = {my(j,v); factorback(Mat(vector(if(n, #n=vecextract(binary(n), "-1..1")), j, [prime(j), n[j]])~))}; \\ From A019565
A319992(n) = { my(m=1); fordiv(n,d,if((dA019565(d))); m; };

a(n) = Product_{d|n, dA019565(d)^[2 == d mod 3].
a(n) = A293214(n) / (A319990(n)*A319991(n)).
For all n >= 1:
A007814(a(n)) = A320005(n).
A048675(a(n)) = A293898(n).
A195017(a(n)) = -A293896(n) mod 3.

A227320 Binary XOR of proper divisors of n.

Original entry on oeis.org

0, 1, 1, 3, 1, 0, 1, 7, 2, 6, 1, 2, 1, 4, 7, 15, 1, 15, 1, 8, 5, 8, 1, 6, 4, 14, 11, 14, 1, 6, 1, 31, 9, 18, 3, 21, 1, 16, 15, 20, 1, 26, 1, 26, 1, 20, 1, 14, 6, 21, 19, 16, 1, 6, 15, 26, 17, 30, 1, 4, 1, 28, 25, 63, 9, 58, 1, 52, 21, 38, 1, 33, 1, 38, 17, 50, 13, 54, 1

Offset: 1

Alex Ratushnyak, Jul 06 2013

An alternative definition (with A027751) would define a(1)=1. - R. J. Mathar, Jul 14 2013

However, this definition is more aligned with A001065 and A218403 where the initial term a(1) is also 0. - Antti Karttunen, Oct 08 2017

Antti Karttunen, Table of n, a(n) for n = 1..16383

Cf. A001065, A027751, A178910, A218403, A248663, A293214, A293215.

Array[BitXor @@ Most@ Divisors@ # &, 79] (* Michael De Vlieger, Oct 08 2017 *)

A227320(n) = { my(s=0); fordiv(n,d,if(dAntti Karttunen, Oct 08 2017

a(n) = A178910(n) XOR n, where XOR is the binary logical exclusive or operator.
From Antti Karttunen, Oct 08 2017: (Start)
a(n) = A248663(A293214(n)).
a(n) <= A218403(n) <= A001065(n).
(End)

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)).

A300834 a(n) = Product_{d|n, dA019565(A003714(d)), where A003714(n) is the n-th Fibbinary number.

Original entry on oeis.org

1, 2, 2, 6, 2, 30, 2, 60, 10, 42, 2, 4200, 2, 126, 70, 660, 2, 9240, 2, 13860, 210, 330, 2, 5082000, 14, 78, 220, 32760, 2, 3783780, 2, 42900, 550, 780, 294, 924924000, 2, 1092, 130, 41621580, 2, 3898440, 2, 112200, 60060, 306, 2, 28078050000, 42, 235620, 1300, 92820, 2, 200119920, 770, 128648520, 1820, 1122, 2, 424964656116000, 2, 3366

Offset: 1

Antti Karttunen, Mar 18 2018

nonn

Antti Karttunen, Table of n, a(n) for n = 1..3194

Cf. A003714, A019565, A300835 (rgs-transform of this sequence), A300836.

Cf. also A293214, A293216, A293221, A293222, A293231.

A072649(n) = { my(m); if(n<1, 0, m=0; until(fibonacci(m)>n, m++); m-2); }; \\ From A072649
A003714(n) = { my(s=0,w); while(n>2, w = A072649(n); s += 2^(w-1); n -= fibonacci(w+1)); (s+n); }
A019565(n) = {my(j,v); factorback(Mat(vector(if(n, #n=vecextract(binary(n), "-1..1")), j, [prime(j), n[j]])~))}; \\ From A019565
A300834(n) = { my(m=1); fordiv(n,d,if(d < n,m *= A019565(A003714(d)))); m; };

a(n) = Product_{d|n, dA019565(A003714(d)).

For n >= 1, A001222(a(n)) = A300836(n).

A318834 a(n) = Product_{d|n, dA019565(phi(d)), where phi is the Euler totient function A000010.

Original entry on oeis.org

1, 2, 2, 4, 2, 12, 2, 12, 6, 20, 2, 108, 2, 60, 30, 60, 2, 540, 2, 300, 90, 84, 2, 2700, 10, 140, 90, 2700, 2, 6300, 2, 420, 126, 44, 150, 121500, 2, 132, 210, 10500, 2, 283500, 2, 5292, 3150, 660, 2, 132300, 30, 5500, 66, 14700, 2, 267300, 210, 472500, 198, 1540, 2, 4630500, 2, 4620, 47250, 4620, 350, 873180, 2, 1452, 990

Offset: 1

Antti Karttunen, Sep 04 2018

nonn

Antti Karttunen, Table of n, a(n) for n = 1..4096

Cf. A000010, A019565, A318835 (rgs-transform).

Cf. also A293214, A293231, A300834.

A019565(n) = {my(j,v); factorback(Mat(vector(if(n, #n=vecextract(binary(n), "-1..1")), j, [prime(j), n[j]])~))}; \\ From A019565
A318834(n) = { my(m=1); fordiv(n,d,if(d < n,m *= A019565(eulerphi(d)))); m; };

a(n) = Product_{d|n, dA019565(A000010(d)).

A048675(a(n)) = A051953(n).

A319990 a(n) = Product_{d|n, dA019565(d)^[0 == d mod 3].

Original entry on oeis.org

1, 1, 1, 1, 1, 6, 1, 1, 6, 1, 1, 90, 1, 1, 6, 1, 1, 1260, 1, 1, 6, 1, 1, 3150, 1, 1, 84, 1, 1, 18900, 1, 1, 6, 1, 1, 1455300, 1, 1, 6, 1, 1, 9900, 1, 1, 17640, 1, 1, 242550, 1, 1, 6, 1, 1, 19209960, 1, 1, 6, 1, 1, 764032500, 1, 1, 9240, 1, 1, 2340, 1, 1, 6, 1, 1, 7283776500, 1, 1, 1260, 1, 1, 35100, 1, 1, 38808, 1, 1, 94594500, 1, 1, 6, 1, 1

Offset: 1

Antti Karttunen, Oct 03 2018

nonn

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

Cf. A293214, A319991, A319992, A320003, A320010 (rgs-transform).

A019565(n) = {my(j,v); factorback(Mat(vector(if(n, #n=vecextract(binary(n), "-1..1")), j, [prime(j), n[j]])~))}; \\ From A019565
A319990(n) = { my(m=1); fordiv(n,d,if((dA019565(d))); m; };

a(n) = Product_{d|n, dA019565(d)^[0 == d mod 3].
a(n) = A293214(n) / (A319991(n)*A319992(n)).
For all n >= 1:
A007814(a(n)) = A320003(n).
A195017(a(n)) = 0 mod 3.

A218403 Bitwise OR of all proper divisors of n; a(1) = 0 by convention.

Original entry on oeis.org

0, 1, 1, 3, 1, 3, 1, 7, 3, 7, 1, 7, 1, 7, 7, 15, 1, 15, 1, 15, 7, 11, 1, 15, 5, 15, 11, 15, 1, 15, 1, 31, 11, 19, 7, 31, 1, 19, 15, 31, 1, 31, 1, 31, 15, 23, 1, 31, 7, 31, 19, 31, 1, 31, 15, 31, 19, 31, 1, 31, 1, 31, 31, 63, 13, 63, 1, 55, 23, 47, 1, 63, 1

Offset: 1

Reinhard Zumkeller, Oct 28 2012

			n=20: properDivisors(20) = {1, 2, 4, 5, 10}, 0001 OR 0010 OR 0100 OR 0101 OR 1010 = 1111 -> a(20) = 15;
n=21: properDivisors(21) = {1, 3, 7}, 001 OR 011 OR 111 = 111 -> a(21) = 7;
n=22: properDivisors(22) = {1, 2, 11}, 0001 OR 0010 OR 1011 = 1111 -> a(22) = 11;
n=23: properDivisors(23) = {1} -> a(23) = 23;
n=24: properDivisors(24) = {1, 2, 3, 4, 6, 8, 12}, 0001 OR 0010 OR 0011 OR 0100 OR 0110 OR 1000 OR 1100 = 1111 -> a(24) = 15;
n=25: properDivisors(25) = {1, 5}, 001 OR 101 = 101 -> a(25) = 5.

Reinhard Zumkeller, Table of n, a(n) for n = 1..8191
Eric Weisstein's World of Mathematics, OR
Wikipedia, Bitwise operation OR
Index entries for sequences related to binary expansion of n

Cf. A001065, A027751, A000225 (subsequence), A123345, A227320, A293214, A293215.

import Data.Bits ((.|.))
a218403 = foldl (.|.)  0 . a027751_row :: Integer -> Integer

Table[BitOr@@Most[Divisors[n]],{n,80}] (* Harvey P. Dale, Nov 09 2012 *)

A218403(n) = { my(s=0); fordiv(n,d,if(dAntti Karttunen, Oct 08 2017

a(n) <= A218388(n).
a(A000040(n)) = 1.
From Antti Karttunen, Oct 08 2017: (Start)
a(n) = A087207(A293214(n)).
A227320(n) <= a(n) <= A001065(n).
(End)

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)).

cp's OEIS Frontend

A300832 a(n) = Product_{d|n} A019565(d)^[moebius(n/d) = -1].

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A319991 a(n) = Product_{d|n, dA019565(d)^[1 == d mod 3].

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A319992 a(n) = Product_{d|n, dA019565(d)^[2 == d mod 3].

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A227320 Binary XOR of proper divisors of n.

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

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A300834 a(n) = Product_{d|n, dA019565(A003714(d)), where A003714(n) is the n-th Fibbinary number.

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A318834 a(n) = Product_{d|n, dA019565(phi(d)), where phi is the Euler totient function A000010.

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A319990 a(n) = Product_{d|n, dA019565(d)^[0 == d mod 3].

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A218403 Bitwise OR of all proper divisors of n; a(1) = 0 by convention.

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