A379234 -id:A379234 - 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

A379236 Numbers k such that x=(sigma(k) XOR 2*k) divides k in carryless binary arithmetic, when the binary expansions of k and x are interpreted as polynomials in ring GF(2)[X].

Original entry on oeis.org

10, 12, 18, 20, 24, 40, 56, 88, 104, 116, 136, 184, 196, 224, 312, 368, 428, 464, 520, 528, 650, 672, 760, 884, 992, 1472, 1504, 1888, 1952, 2528, 3424, 3724, 4832, 5312, 6464, 7136, 9112, 11096, 11288, 11744, 13216, 15352, 15376, 15872, 15968, 16256, 17816, 17964, 22616, 24448, 26728, 28544, 29296, 30592, 30656

Offset: 1

Antti Karttunen, Jan 05 2025

nonn

Among the first 484 terms, there are no odd numbers, the only squares are 196, 15376, 1032256, and 18 is the only twice square.

			196 is a term as sigma(196) = 399, 2*196 XOR 399 = 7 is not zero, and A048720(7, 89) = 399.

Cf. A000203, A000396, A003987, A048720, A280500, A318467.
Cf. A379234 (subsequence).
Cf. also A097498 (= A153501 U A271816).

divides_in_GF2X(a,b) = { my(Pa=Pol(binary(a))*Mod(1, 2), Pb=Pol(binary(b))*Mod(1, 2)); !lift(Pa % Pb); };
is_A379236(n) = { my(s=sigma(n), x=bitxor(2*n, s)); (x && divides_in_GF2X(n, x)); };

{k such that k = A048720(A318467(k), x) for some x > 0}.

{k not in A000396 such that A280500(k, A318467(k)) > 0}.

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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