A378227 -id:A378227 - cp's OEIS frontend

A318467 a(n) = 2*n XOR A000203(n), where XOR is bitwise-xor (A003987) and A000203 = sum of divisors.

3, 7, 2, 15, 12, 0, 6, 31, 31, 6, 26, 4, 20, 4, 6, 63, 48, 3, 50, 2, 10, 8, 54, 12, 45, 30, 30, 0, 36, 116, 30, 127, 114, 114, 118, 19, 108, 112, 118, 10, 120, 52, 122, 12, 20, 20, 110, 28, 91, 57, 46, 10, 92, 20, 38, 8, 34, 46, 74, 208, 68, 28, 22, 255, 214, 20, 194, 246, 234, 28, 198, 83, 216, 230, 234, 20, 250, 52, 206, 26

Offset: 1

Antti Karttunen, Aug 26 2018

Cf. A000203, A003987, A318457, A318458, A318466, A318468.
Cf. A000396 (positions of zeros), A378227 (XOR-Moebius transform), A379234 (fixed points), A379236.
Cf. also A294899, A318457, A378988.

Table[BitXor[2n,DivisorSigma[1,n]],{n,80}] (* Harvey P. Dale, Oct 30 2022 *)

PARI
```
A318467(n) = bitxor(2*n,sigma(n));
```

a(n) = A003987(2*n, A000203(n)).
a(n) = A224880(n) - 2*A318468(n).
a(n) = 2*n XOR (A318457(n)+2*A318458(n)). - Antti Karttunen, Jan 08 2025

A378226 XOR-Moebius transform of A318457, where A318457(n) = n XOR (sigma(n)-n).

Original entry on oeis.org

1, 2, 3, 4, 5, 0, 7, 8, 15, 4, 11, 24, 13, 0, 1, 16, 17, 8, 19, 4, 27, 16, 23, 40, 27, 4, 27, 0, 29, 52, 31, 32, 39, 36, 45, 8, 37, 32, 57, 16, 41, 0, 43, 24, 5, 32, 47, 80, 63, 0, 53, 20, 53, 104, 41, 112, 63, 4, 59, 124, 61, 0, 7, 64, 91, 48, 67, 76, 75, 36, 71, 0, 73, 68, 103, 56, 83, 36, 79, 48, 111, 84, 83

Offset: 1

Antti Karttunen, Nov 26 2024

nonn

Unique sequence satisfying SumXOR_{d divides n} a(d) = A318457(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.

Cf. A000203, A001065, A003987, A318457, A378227, A378230 (positions of 0's), A378441 (fixed points).

A318457(n) = bitxor(n,sigma(n)-n);
A378226(n) = { my(v=0); fordiv(n, d, if(issquarefree(n/d), v=bitxor(v, A318457(d)))); (v); }

cp's OEIS Frontend

A318467 a(n) = 2*n XOR A000203(n), where XOR is bitwise-xor (A003987) and A000203 = sum of divisors.

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