A295901 -id:A295901 - cp's OEIS frontend

A324712 Xor-Moebius transform of A323243, where A323243(n) = sigma(A156552(n)).

0, 1, 3, 5, 7, 4, 15, 12, 15, 11, 31, 15, 63, 28, 22, 16, 127, 7, 255, 28, 43, 46, 511, 24, 27, 106, 20, 39, 1023, 60, 2047, 56, 42, 206, 34, 44, 4095, 508, 172, 32, 8191, 24, 16383, 113, 57, 734, 32767, 56, 51, 9, 128, 213, 65535, 4, 67, 116, 748, 1768, 131071, 26, 262143, 3406, 20, 72, 70, 125, 524287, 389, 762, 22

Offset: 1

Antti Karttunen, Mar 12 2019

nonn

Properties of Xor-Moebius transform are explained in A295901.

Antti Karttunen, Table of n, a(n) for n = 1..10000 (based on Hans Havermann's factorization of A156552)
Index entries for sequences related to binary expansion of n
Index entries for sequences computed from indices in prime factorization
Index entries for sequences related to sigma(n)

Cf. A156552, A295901, A297106, A323243, A324543, A324713, A324714.

A324712(n) = { my(v=0); fordiv(n, d, if(issquarefree(n/d), v=bitxor(v, A323243(d)))); (v); } \\ For A323243, see under that entry.

a(n) = A297106(n) XOR A324714(n).

a(A000040(n)) = A000225(n).

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)

A297106 Xor-Moebius transform of A156552.

Original entry on oeis.org

0, 1, 2, 2, 4, 6, 8, 4, 4, 12, 16, 12, 32, 24, 12, 8, 64, 12, 128, 24, 24, 48, 256, 24, 8, 96, 8, 48, 512, 20, 1024, 16, 48, 192, 24, 24, 2048, 384, 96, 48, 4096, 40, 8192, 96, 24, 768, 16384, 48, 16, 24, 192, 192, 32768, 24, 48, 96, 384, 1536, 65536, 40, 131072, 3072, 48, 32, 96, 80, 262144, 384, 768, 40, 524288, 48

Offset: 1

Antti Karttunen, Dec 25 2017

nonn

Unique sequence satisfying SumXOR_{d divides n} a(d) = A156552(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 the Xor-Moebius transform.
The ordinary Möbius transform of A156552 is given in A297112.
It seems that A091629 gives the fixed points of this sequence.

Antti Karttunen, Table of n, a(n) for n = 1..2048

Cf. A064989, A091629, A156552, A295901, A297108, A297112.

A064989(n) = {my(f); f = factor(n); if((n>1 && f[1,1]==2), f[1,2] = 0); for (i=1, #f~, f[i,1] = precprime(f[i,1]-1)); factorback(f)};
A156552(n) = if(1==n, 0, if(!(n%2), 1+(2*A156552(n/2)), 2*A156552(A064989(n))));
A297106(n) = { my(v=0); fordiv(n, d, if(issquarefree(n/d), v=bitxor(v, A156552(d)))); (v); } \\ after code in A295901.

A297108 If n is prime(k)^e, e >= 1, then a(n) = 2^(k-1), otherwise 0; Möbius transform of A048675.

Original entry on oeis.org

0, 1, 2, 1, 4, 0, 8, 1, 2, 0, 16, 0, 32, 0, 0, 1, 64, 0, 128, 0, 0, 0, 256, 0, 4, 0, 2, 0, 512, 0, 1024, 1, 0, 0, 0, 0, 2048, 0, 0, 0, 4096, 0, 8192, 0, 0, 0, 16384, 0, 8, 0, 0, 0, 32768, 0, 0, 0, 0, 0, 65536, 0, 131072, 0, 0, 1, 0, 0, 262144, 0, 0, 0, 524288, 0, 1048576, 0, 0, 0, 0, 0, 2097152, 0, 2, 0, 4194304

Offset: 1

Antti Karttunen, Dec 25 2017

This is also Xor-Moebius transform of A248663, in other words, the unique sequence satisfying SumXOR_{d divides n} a(d) = A248663(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 this transform.

Antti Karttunen, Table of n, a(n) for n = 1..1024

Cf. A008683, A048675, A248663, A295901, A297106, A297109.

A297108(n) = if(1==omega(n),2^(primepi(factor(n)[1,1])-1),0);
\\ A more complicated way which demonstrates the Moebius transform:
A048675(n) = { my(f = factor(n)); sum(k=1, #f~, f[k, 2]*2^primepi(f[k, 1]))/2; }; \\ This function after Michel Marcus
A297108(n) = sumdiv(n,d,moebius(n/d)*A048675(d));
\\ And yet another way demonstrating the comment:
A248663(n) = A048675(core(n));
A297108(n) = { my(v=0); fordiv(n, d, if(issquarefree(n/d), v=bitxor(v, A248663(d)))); (v); } \\ after code in A295901.

If A001221(n) = 1 [when n is in A000961], then a(n) = 2^(A297109(n)-1) = 2^(A055396(n)-1), otherwise 0.

a(n) = Sum_{d|n} A048675(d)*A008683(n/d).

A296203 Xor-Moebius transform of A000203, the sum of divisors.

Original entry on oeis.org

1, 2, 5, 4, 7, 10, 9, 8, 9, 22, 13, 20, 15, 18, 27, 16, 19, 34, 21, 60, 45, 42, 25, 40, 25, 38, 37, 36, 31, 78, 33, 32, 57, 38, 63, 108, 39, 42, 51, 120, 43, 90, 45, 116, 95, 82, 49, 80, 49, 86, 95, 76, 55, 122, 67, 72, 65, 70, 61, 204, 63, 66, 65, 64, 93, 130, 69, 76, 125, 166, 73, 184, 75, 86, 125, 180, 101, 190, 81, 240, 81

Offset: 1

Antti Karttunen, Dec 25 2017, from Rémy Sigrist's "gallery of scatter plots" in A295901

nonn

Unique sequence satisfying SumXOR_{d divides n} a(d) = sigma(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
Rémy Sigrist, Scatterplot of the first 2^16 terms

Cf. A000203, A169813, A256739, A295901, A296208.

A296203(n) = { my(v=0); fordiv(n, d, if(issquarefree(n/d), v=bitxor(v, sigma(d)))); (v); } \\ after code in A295901.

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

Original entry on oeis.org

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).

A324714 a(n) = A297106(n) XOR A324712(n).

Original entry on oeis.org

0, 0, 1, 7, 3, 2, 7, 8, 11, 7, 15, 3, 31, 4, 26, 24, 63, 11, 127, 4, 51, 30, 255, 0, 19, 10, 28, 23, 511, 40, 1023, 40, 26, 14, 58, 52, 2047, 124, 204, 16, 4095, 48, 8191, 17, 33, 478, 16383, 8, 35, 17, 64, 21, 32767, 28, 115, 20, 876, 232, 65535, 50, 131071, 334, 36, 104, 38, 45, 262143, 5, 506, 62, 524287, 68, 1048575, 1786, 59

Offset: 1

Antti Karttunen, Mar 13 2019

nonn

a(n) is also the XOR-Moebius transform of (A156552(d) XOR A323243(d)). See A295901 for the properties of that transform.

Antti Karttunen, Table of n, a(n) for n = 1..10000 (based on Hans Havermann's factorization of A156552)
Index entries for sequences related to binary expansion of n
Index entries for sequences computed from indices in prime factorization
Index entries for sequences related to sigma(n)

Cf. A156552, A295901, A297106, A323243, A324712, A324713.

A324714(n) = bitxor(A324712(n), A297106(n));

A324714(n) = { my(v=0); fordiv(n, d, if(issquarefree(n/d), v=bitxor(v, bitxor(A323243(d),A156552(d))))); (v); }

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.

A296208 Xor-Moebius transform of A005187.

Original entry on oeis.org

1, 2, 5, 4, 9, 12, 10, 8, 20, 24, 18, 24, 22, 16, 23, 16, 33, 60, 34, 48, 41, 56, 43, 48, 39, 36, 34, 40, 55, 52, 56, 32, 86, 96, 65, 120, 70, 104, 88, 96, 78, 104, 83, 120, 88, 112, 88, 96, 84, 84, 71, 80, 103, 104, 115, 80, 72, 68, 112, 96, 116, 76, 75, 64, 158, 244, 130, 192, 168, 192, 139, 240, 142, 212, 175, 216

Offset: 1

Antti Karttunen, Dec 25 2017

Unique sequence satisfying SumXOR_{d divides n} a(d) = A005187(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. A297111 gives the ordinary Möbius transform of A005187.

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

Cf. A003987, A005187, A256739, A294899, A295901, A296203, A297110, A297111.

A005187(n) = { my(s=n); while(n>>=1, s+=n); s; };
A296208(n) = { my(v=0); fordiv(n, d, if(issquarefree(n/d), v=bitxor(v, A005187(d)))); (v); } \\ after code in A295901.

A297110 Xor-Moebius transform of A006068, inverse of the binary Gray code.

Original entry on oeis.org

1, 2, 3, 4, 7, 4, 4, 8, 12, 8, 12, 8, 8, 12, 15, 16, 31, 20, 28, 16, 31, 20, 27, 16, 23, 24, 28, 24, 23, 16, 20, 32, 48, 32, 63, 40, 56, 36, 48, 32, 48, 32, 51, 40, 48, 44, 52, 32, 36, 56, 63, 48, 39, 36, 47, 48, 48, 56, 44, 32, 40, 60, 63, 64, 112, 80, 124, 64, 96, 64, 123, 80, 112, 72, 111, 72, 127, 80, 116, 64

Offset: 1

Antti Karttunen, Dec 25 2017

Unique sequence satisfying SumXOR_{d divides n} a(d) = A006068(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 the 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. A006068, A256739, A295901, A296208.

A006068(n)= { my(s=1, ns); while(1, ns = n >> s; if(0==ns, break()); n = bitxor(n, ns); s <<= 1; ); return (n); } \\ Essentially Joerg Arndt's Jul 19 2012 code.
A297110(n) = { my(v=0); fordiv(n, d, if(issquarefree(n/d), v=bitxor(v, A006068(d)))); (v); };

cp's OEIS Frontend

A324712 Xor-Moebius transform of A323243, where A323243(n) = sigma(A156552(n)).

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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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A297106 Xor-Moebius transform of A156552.

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A297108 If n is prime(k)^e, e >= 1, then a(n) = 2^(k-1), otherwise 0; Möbius transform of A048675.

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A296203 Xor-Moebius transform of A000203, the sum of divisors.

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

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A324714 a(n) = A297106(n) XOR A324712(n).

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A296206 Xor-Moebius transform applied twice to natural numbers.

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