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A339599
Quadratically perfect numbers: numbers k such that (sigma(k) - 2k)^2 = sigma(k).
0
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
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
KEYWORD
nonn,more
AUTHOR
Tomohiro Yamada, Dec 09 2020
STATUS
approved