A379236 -id:A379236 - 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

A097498 Numbers k divisible by their abundance sigma(k) - 2*k.

Original entry on oeis.org

1, 2, 4, 8, 10, 12, 16, 18, 20, 24, 32, 40, 44, 56, 64, 88, 104, 120, 128, 136, 152, 184, 196, 224, 234, 256, 368, 464, 512, 650, 672, 752, 884, 992, 1024, 1504, 1888, 1952, 2048, 2144, 2272, 2528, 3724, 4096, 5624, 8192, 8384, 9112, 11096, 12224, 13736

Offset: 1

Reinhard Zumkeller, Aug 24 2004

nonn

Amiram Eldar, Table of n, a(n) for n = 1..417 (terms 1..348 from Charles R Greathouse IV)
Eric Weisstein's World of Mathematics, Abundance.

Cf. A000079 (subsequence), A033879, A033880.
Disjoint union of A153501 and A271816.
Cf. also A379236.

Select[Range[15000],Divisible[#,DivisorSigma[1,#]-2#]&]//Quiet (* Harvey P. Dale, Mar 15 2018 *)

is(n)=my(t=sigma(n)-2*n); t && n%t==0 \\ Charles R Greathouse IV, Dec 12 2014

A379234 Numbers k for which k XOR 2*k = sigma(k), where sigma is the sum of divisors function.

Original entry on oeis.org

312, 428, 672, 760, 5009850

Offset: 1

Antti Karttunen, Jan 04 2025

Equally, numbers k such that 2*k XOR sigma(k) = k, i.e., k XOR sigma(k) = 2*k.

If it exists, a(6) > 2^33.

			672 has binary expansion 1010100000_2, and 672 XOR 2*672 has binary expansion 11111100000_2 = 2016 (= 63*32) = sigma(672), so 672 is included in this sequence. Notably, as 672 is also a Fibbinary number (in A003714, no adjacent 1-bits), it follows that 672 XOR 2*672 = 3*672, and thus 672 is also a 3-perfect number, A005820.

Cf. A000203, A003714, A003987, A005820, A048724.
Fixed points of A318467.
Subsequence of A379236.

is_A379234(n) = (bitxor(2*n,n)==sigma(n));

{k such that A000203(k) = A048724(k)}.

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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A097498 Numbers k divisible by their abundance sigma(k) - 2*k.

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A379234 Numbers k for which k XOR 2*k = sigma(k), where sigma is the sum of divisors function.

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