OFFSET
1,1
COMMENTS
Integers k that have a divisor d such that sigma(k) - d - k/d = 2*k.
Note that this is not necessarily the same as just being the numbers that are strongly pseudoperfect and also 2-near perfect. This is because a number might be strongly pseudoperfect for one set of divisors which requires more than one redundant pair, while also being 2-near perfect due to removing a different pair. (This probably never actually happens.) - Joshua Zelinsky, Nov 09 2023
LINKS
Vedant Aryan, Dev Madhavani, Savan Parikh, Ingrid Slattery, and Joshua Zelinsky, On 2-Near Perfect Numbers, arXiv:2310.01305 [math.NT], 2023. See p. 13.
EXAMPLE
156 is strongly 2-near perfect since sigma(156) = 392, 2*78 = 156, and 392-2-78 = 2*156.
MATHEMATICA
fQ[n_]:=AnyTrue[Table[DivisorSigma[1, n]-Divisors[n][[i]]-n/Divisors[n][[i]], {i, DivisorSigma[0, n]}], #==2*n&]; Select[Range[61000], fQ[#]&] (* Ivan N. Ianakiev, Oct 04 2023 *)
PROG
(PARI) isok(k) = my(s=sigma(k)); fordiv(k, d, if (s-d-k/d == 2*k, return(1)));
CROSSREFS
KEYWORD
nonn
AUTHOR
Michel Marcus, Oct 03 2023
EXTENSIONS
a(21) from Ivan N. Ianakiev, Oct 04 2023
a(22)-a(30) from Amiram Eldar, Sep 20 2024
STATUS
approved