A296207 -id:A296207 - cp's OEIS frontend

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.

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.

A256739 Unique sequence satisfying SumXOR_{d divides n} a(d) = n for any n>0, where SumXOR is the analog of summation under the binary XOR operation.

Original entry on oeis.org

1, 3, 2, 6, 4, 6, 6, 12, 10, 12, 10, 12, 12, 10, 8, 24, 16, 30, 18, 24, 16, 30, 22, 24, 28, 20, 18, 20, 28, 24, 30, 48, 40, 48, 32, 60, 36, 54, 40, 48, 40, 48, 42, 60, 40, 58, 46, 48, 54, 36, 32, 40, 52, 54, 56, 40, 40, 36, 58, 48, 60, 34, 32, 96, 72, 120, 66

Offset: 1

Paul Tek, Apr 09 2015

Replacing "SumXOR" by "Sum" in the name leads to the Euler totient function (A000010).
Replacing "SumXOR" by "Product" in the name leads to the exponential of Mangoldt function (A014963).
a(p) = p-1 for any prime p>2.
a(2^k) = 2^k+2^(k-1) for any k>0.
A070939(a(n)) = A070939(n) for any n>0.
The graph of this sequence is quite remarkable. - N. J. A. Sloane, Apr 09 2015
Xor-Moebius transform of natural numbers, A000027. See A295901 for a list of some of the properties of this transform. - Antti Karttunen, Dec 29 2017

Paul Tek, Table of n, a(n) for n = 1..16383
Paul Tek, PARI program for this sequence

Cf. A000010, A003987, A014963, A295901, A296206, A296207, A297107 (fixed points).

a = Table[0, {16383}];
Do[pa = n; Do[pa = BitXor[pa, a[[d]]], {d, Divisors[n]}]; a[[n]] = pa, {n, Length[a]}];
a (* Jean-François Alcover, Oct 18 2019, after Paul Tek *)

PARI
```
\\ See Links section.
```

A256739(n) = { my(v=0); fordiv(n, d, if(issquarefree(n/d), v=bitxor(v, d))); (v); } \\ Antti Karttunen, Dec 29 2017, after code in A295901.

a(n) = n XOR ( SumXOR_{d divides n and d < n} a(d) ) for any n>0.
From Antti Karttunen, Dec 29 2017: (Start)
a(n) = SumXOR_{d|n} A296206(d).
a(n) = n XOR A296207(n), where XOR is bitwise exclusive or, A003987.
(End)

A296206 Xor-Moebius transform applied twice to natural numbers.

Original entry on oeis.org

1, 2, 3, 5, 5, 6, 7, 10, 8, 10, 11, 15, 13, 14, 15, 20, 17, 16, 19, 17, 21, 22, 23, 30, 24, 26, 24, 27, 29, 30, 31, 40, 33, 34, 35, 40, 37, 38, 39, 34, 41, 42, 43, 39, 40, 46, 47, 60, 48, 48, 51, 57, 53, 48, 55, 54, 57, 58, 59, 51, 61, 62, 56, 80, 65, 66, 67, 85, 69, 70, 71, 80, 73, 74, 72, 95, 77, 78, 79, 68, 88

Offset: 1

Antti Karttunen, Dec 25 2017

Xor-Moebius transform of A256739, in other words, the unique sequence that satisfies SumXOR_{d divides n} a(d) = A256739(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, A256739, A295901, A296207.

A256739(n) = { if(1==n,return(n)); my(x=n); fordiv(n,d,if(dA256739(d)))); (x); };
A296206(n) = { my(v=0); fordiv(n, d, if(issquarefree(n/d), v=bitxor(v, A256739(d)))); (v); } \\ after code in A295901.

A297107 Fixed points of A256739, Xor-Moebius transform of natural numbers.

Original entry on oeis.org

1, 6, 12, 24, 48, 54, 96, 108, 192, 216, 384, 432, 486, 768, 864, 972, 1536, 1728, 1944, 3072, 3456, 3888, 6144, 6912, 7776, 12288, 13824, 15552, 24576, 27648, 31104, 49152, 55296, 62208, 98304, 110592, 124416, 196608, 221184, 248832, 393216, 442368, 497664, 786432, 884736, 995328, 1572864, 1769472, 1990656, 3145728, 3538944, 3981312

Offset: 1

Antti Karttunen, Dec 25 2017

nonn

After the initial 1, all terms seem to be multiples of six.

Fixed points of A256739. Positions of zeros in A296207.

cp's OEIS Frontend

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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A256739 Unique sequence satisfying SumXOR_{d divides n} a(d) = n for any n>0, where SumXOR is the analog of summation under the binary XOR operation.

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A296206 Xor-Moebius transform applied twice to natural numbers.

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PARI

A297107 Fixed points of A256739, Xor-Moebius transform of natural numbers.

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