cp's OEIS Frontend

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

%I A339599 #14 Dec 29 2020 07:07:28
%S A339599 1,3,66,491536
%N A339599 Quadratically perfect numbers: numbers k such that (sigma(k) - 2k)^2 = sigma(k).
%C A339599 Paul Lescot observed that if m is a perfect number and 2m - 1 is prime, then m * (2m - 1) belongs to this sequence.
%C A339599 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.
%C A339599 An even term k in this sequence belongs to A256151 since k = m * (2m - 1) and sigma(k) is square.
%C A339599 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.
%C A339599 There exists no further term in this sequence in n <= 2^32.
%C A339599 Question: are there only four terms in this sequence?
%H A339599 Paul Lescot, <a href="https://doi.org/10.1017/mag.2020.3">An arithmetical question related to perfect numbers</a>, The Mathematical Gazette 104 (2020), 20-26, a preprint available from <a href="https://hal.ird.fr/LMRS/hal-02050720v1">Archive ouverte HAL</a>.
%e A339599 sigma(1) = 1 = (sigma(1) - 2)^2;
%e A339599 sigma(3) = 4 = (sigma(3) - 6)^2;
%e A339599 sigma(66) = 144 = 12^2 = (sigma(66) - 132)^2.
%o A339599 (PARI) is(k) = {sigma(k) == (sigma(k)-2*k)^2}
%Y A339599 Cf. A000396, A256151, A033880.
%K A339599 nonn,more
%O A339599 1,2
%A A339599 _Tomohiro Yamada_, Dec 09 2020