A366172 Strongly 2-near perfect numbers.

156, 352, 6832, 60976, 91648, 152812, 260865, 2834572, 3335968, 3532096, 4077388, 5044725, 5725504, 6112576, 8102656, 10557148, 19762876, 39411712, 50718016, 66965104, 111372508, 232774912, 483879808, 2045453824, 6849461025, 7904670976, 8521265152, 11818720108, 13112466688, 13714642432

Offset: 1

Author

Michel Marcus, Oct 03 2023

Keywords

• nonn

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

Examples

			156 is strongly 2-near perfect since sigma(156) = 392, 2*78 = 156, and 392-2-78 = 2*156.
		

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.

Crossrefs

Cf. A000203 (sigma).

Subsequence of A005835.

Intersection of A341475 and A334405.

Programs

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 *)

PARI

isok(k) = my(s=sigma(k)); fordiv(k, d, if (s-d-k/d == 2*k, return(1)));

Extensions

a(21) from Ivan N. Ianakiev, Oct 04 2023

a(22)-a(30) from Amiram Eldar, Sep 20 2024