A339599 Quadratically perfect numbers: numbers k such that (sigma(k) - 2k)^2 = sigma(k).

1, 3, 66, 491536

Offset

1,2

Comments

Paul Lescot observed that if m is a perfect number and 2m - 1 is prime, then m * (2m - 1) belongs to this sequence.

Moreover, Lescot proved that an even integer k belongs to this sequences if and only if k = m * (2m - 1) for an even perfect number m = 2^(p-1) * (2^p - 1) and 2m - 1 is prime. Among all primes p in A000043 (Mersenne exponents) up to 23209, only p = 2 and 5, which give 66 and 491536 respectively, satisfy this condition.

An even term k in this sequence belongs to A256151 since k = m * (2m - 1) and sigma(k) is square.

Lescot also confirms that there exists no odd integer between 5 and 10^6 in this sequence and conjectures that there exists no further odd term in this sequence.

There exists no further term in this sequence in n <= 2^32.

Question: are there only four terms in this sequence?

Links

Table of n, a(n) for n=1..4.

Paul Lescot, An arithmetical question related to perfect numbers, The Mathematical Gazette 104 (2020), 20-26, a preprint available from Archive ouverte HAL.

Example

sigma(1) = 1 = (sigma(1) - 2)^2;
sigma(3) = 4 = (sigma(3) - 6)^2;
sigma(66) = 144 = 12^2 = (sigma(66) - 132)^2.

Prog

(PARI) is(k) = {sigma(k) == (sigma(k)-2*k)^2}

Crossrefs

Cf. A000396, A256151, A033880.

Sequence in context: A185053 A165655 A157576 * A105443 A065425 A096482

Adjacent sequences: A339596 A339597 A339598 * A339600 A339601 A339602

Keyword

nonn,more

Author

Tomohiro Yamada, Dec 09 2020

Status

approved