A227320 -id:A227320 - cp's OEIS frontend

A296207 Xor-Moebius transform of A227320, binary XOR of proper divisors of n.

0, 1, 1, 2, 1, 0, 1, 4, 3, 6, 1, 0, 1, 4, 7, 8, 1, 12, 1, 12, 5, 8, 1, 0, 5, 14, 9, 8, 1, 6, 1, 16, 9, 18, 3, 24, 1, 16, 15, 24, 1, 26, 1, 16, 5, 20, 1, 0, 7, 22, 19, 28, 1, 0, 15, 16, 17, 30, 1, 12, 1, 28, 31, 32, 9, 58, 1, 36, 21, 38, 1, 48, 1, 38, 19, 32, 13, 54, 1, 48, 27, 42, 1, 52, 21, 40, 31, 32, 1, 34, 11, 40, 29, 44, 23, 0, 1

Offset: 1

Antti Karttunen, Dec 25 2017

Unique sequence satisfying SumXOR_{d divides n} a(d) = A227320(n) for all n > 0, where SumXOR is the analog of summation under the binary XOR operation. See A295901 for a list of some of the properties of Xor-Moebius transform.

Antti Karttunen, Table of n, a(n) for n = 1..16384
Index entries for sequences related to binary expansion of n

Cf. A003987, A178910, A227320, A256739, A295901, A296206.

Cf. A297107 (positions of zeros).

A227320(n) = { my(s=0); fordiv(n,d,if(dA296207(n) = { my(v=0); fordiv(n, d, if(issquarefree(n/d), v=bitxor(v, A227320(d)))); (v); } \\ after code in A295901.

a(n) = n XOR A256739(n), where XOR is the bitwise XOR-operation (A003987).

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).

A295901 Unique sequence satisfying SumXOR_{d divides n} a(d) = n^2 for any n > 0, where SumXOR is the analog of summation under the binary XOR operation.

Original entry on oeis.org

1, 5, 8, 20, 24, 40, 48, 80, 88, 120, 120, 160, 168, 240, 240, 320, 288, 312, 360, 480, 384, 408, 528, 640, 616, 520, 648, 960, 840, 816, 960, 1280, 1072, 1440, 1248, 1248, 1368, 1224, 1360, 1920, 1680, 1920, 1848, 1632, 1872, 2640, 2208, 2560, 2384, 3016

Offset: 1

Rémy Sigrist, Nov 29 2017

This sequence is a variant of A256739; both sequences have nice graphical features.
Replacing "SumXOR" by "Sum" in the name leads to the Jordan function J_2 (A007434).
For any sequence f of nonnegative integers with positive indices:
- let x_f be the unique sequence satisfying SumXOR_{d divides n} x_f(d) = f(n) for any n > 0,
- in particular, x_A000027 = A256739 and x_A000290 = a (this sequence),
- also, x_A178910 = A000027 and x_A055895 = A000079,
- see the links section for a gallery of x_f plots for some classic f functions,
- x_f(1) = f(1),
- x_f(p) = f(1) XOR f(p) for any prime p,
- x_A063524 = A008966,
- x_f(n) = SumXOR_{d divides n and n/d is squarefree} f(d) for any n > 0,
- the function x: f -> x_f is a bijection,
- A000004 is the only fixed point of x (i.e. x_f = f if and only if f = A000004),
- for any sequence f, x_{2*f} = 2 * x_f,
- for any sequences g and f, x_{g XOR f} = x_g XOR x_f.
From Antti Karttunen, Dec 29 2017: (Start)
The transform x_f described above could be called "Xor-Moebius transform of f" because of its analogous construction to Möbius transform with A008683 replaced by A008966 and the summation replaced by cumulative XOR.
(End)

Rémy Sigrist, Table of n, a(n) for n = 1..16383
Rémy Sigrist, Colored scatterplot of the first 2^17-1 terms (where the color is function of A087207(n) % 8)
Rémy Sigrist, Scatterplot of the first 2^16 terms of x_A000010 (Euler totient function)
Rémy Sigrist, Scatterplot of the first 2^16 terms of x_A000203 (sigma)
Rémy Sigrist, Scatterplot of the first 2^16 terms of x_A001157 (sigma_2)
Rémy Sigrist, Scatterplot of the first 2^16 terms of x_A000040 (prime numbers)
Rémy Sigrist, Scatterplot of the first 2^16 terms of x_A000720 (PrimePi)
Rémy Sigrist, Scatterplot of the first 2^16 terms of x_A006370 (Collatz map)
Rémy Sigrist, Scatterplot of the first 2^16 terms of x_A005132 (Recamán's sequence)

Cf. A000004, A000010, A000027, A000040, A000079, A000290, A000720, A001157, A003987, A005132, A006370, A007434, A008966, A055895, A063524, A087207, A178910.

Same transform applied to other sequences: A256739 (A000027), A296206 (A256739), A296207 (A227320), A296203 (A000203), A296208 (A005187), A297110 (A006068), A297108 (A248663), A297106 (A156552).

a(n{, f=k->k^2}) = my (v=0); fordiv(n,d,if (issquarefree(n/d), v=bitxor(v,f(d)))); return (v)

a(n) = SumXOR_{d divides n and n/d is squarefree} d^2.

A169813 a(n) = n XOR sigma(n), where sigma(n) is the number of divisors of n, A000203.

Original entry on oeis.org

0, 1, 7, 3, 3, 10, 15, 7, 4, 24, 7, 16, 3, 22, 23, 15, 3, 53, 7, 62, 53, 50, 15, 36, 6, 48, 51, 36, 3, 86, 63, 31, 17, 20, 19, 127, 3, 26, 31, 114, 3, 74, 7, 120, 99, 102, 31, 76, 8, 111, 123, 86, 3, 78, 127, 64, 105, 96, 7, 148, 3, 94, 87, 63, 21, 210, 7, 58, 37, 214, 15, 139, 3, 56

Offset: 1

N. J. A. Sloane, May 28 2010

Cf. A000203, A003987, A070883, A169810, A169811, A169812, A169814.

Cf. also A106409, A178910, A227320, A294899.

a(n) = bitxor(n, sigma(n)); \\ Michel Marcus, Nov 25 2017

(define (A169813 n) (A003987bi n (A000203 n))) ;; Where A003987bi implements the bitwise-XOR, A003987 and code for A000203 can be found under that entry. - Antti Karttunen, Nov 25 2017

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)

A318504 SumXOR of divisors of n, up to, but not including the second largest of them A032742(n); a(1) = 0 by convention.

Original entry on oeis.org

0, 0, 0, 1, 0, 3, 0, 3, 1, 3, 0, 4, 0, 3, 2, 7, 0, 6, 0, 2, 2, 3, 0, 10, 1, 3, 2, 0, 0, 9, 0, 15, 2, 3, 4, 7, 0, 3, 2, 0, 0, 15, 0, 12, 14, 3, 0, 22, 1, 12, 2, 10, 0, 29, 4, 6, 2, 3, 0, 26, 0, 3, 12, 31, 4, 27, 0, 22, 2, 5, 0, 5, 0, 3, 8, 20, 6, 17, 0, 4, 11, 3, 0, 14, 4, 3, 2, 18, 0, 3, 6, 16, 2, 3, 4, 46, 0, 10, 0, 5, 0, 53, 0, 24, 26

Offset: 1

Antti Karttunen, Aug 28 2018

Antti Karttunen, Table of n, a(n) for n = 1..65537
Index entries for sequences related to binary expansion of n

Cf. A003987, A032742, A106409, A178910, A227320, A318505, A318507, A318517.

A032742(n) = if(1==n,n,n/vecmin(factor(n)[,1]));
A318504(n) = { my(v=0); fordiv(n,d,if(d<A032742(n), v = bitxor(v,d))); (v); };

a(n) = A032742(n) XOR A227320(n).

For n > 1, a(n) = A106409(n) XOR A178910(n).

cp's OEIS Frontend

A296207 Xor-Moebius transform of A227320, binary XOR of proper divisors of n.

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

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A295901 Unique sequence satisfying SumXOR_{d divides n} a(d) = n^2 for any n > 0, where SumXOR is the analog of summation under the binary XOR operation.

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A169813 a(n) = n XOR sigma(n), where sigma(n) is the number of divisors of n, A000203.

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

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A318504 SumXOR of divisors of n, up to, but not including the second largest of them A032742(n); a(1) = 0 by convention.

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