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

312, 428, 672, 760, 5009850

Offset: 1

Author

Antti Karttunen, Jan 04 2025

Keywords

• nonn
• hard
• more

Comments

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.

Examples

			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.
		

Links

• Index entries for sequences related to binary expansion of n.
• Index entries for sequences related to sigma(n).

Crossrefs

Cf. A000203, A003714, A003987, A005820, A048724.

Fixed points of A318467.

Subsequence of A379236.

Programs

PARI

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

Formula

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