OFFSET
1,1
COMMENTS
f-perfect numbers are defined in A066218.
m is in the sequence iff sigma(m) = 2*m - d(m) + 2, where d = A000005 is the number of divisors. If 2^k + 2*k - 3 is prime (k in A392535), then 2^(k-1)*(2^k + 2*k - 3) is a term. - Farideh Firoozbakht, Sep 18 2006; updated by Max Alekseyev, Jan 15 2026
Also, numbers n such that the number of nontrivial proper subgroups of a dihedral group D_{2n} is the same as its order. - Ivan Neretin, Jun 21 2016, after Dietrich Burde, see MSE link
LINKS
Max A. Alekseyev, Computing bounded solutions to linear Diophantine equations with the sum of divisors, arXiv:2601.17832 [math.NT], 2026. See p. 10.
Joseph L. Pe, On a Generalization of Perfect Numbers, J. Rec. Math., 31(3) (2002-2003), 168-172.
Math StackExchange, Discussion on math.SE.
EXAMPLE
f(10) = 11 = 2 + 3 + 6 = f(1) + f(2) + f(5), hence 10 is a term of the sequence.
MATHEMATICA
Select[ Range[ 500000 ], DivisorSigma[ 1, # ] == 2# - DivisorSigma[ 0, # ] + 2 & ] (* Farideh Firoozbakht, Sep 18 2006 *)
f[x_] := x + 1; Select[ Range[ 1, 10^5], 2 * f[ # ] == Apply[ Plus, Map[ f, Divisors[ # ] ] ] & ]
PROG
(PARI) isok(m) = sigma(m) == 2*m-numdiv(m)+2; \\ Michel Marcus, Mar 13 2020
CROSSREFS
KEYWORD
nonn,more
AUTHOR
Joseph L. Pe, Dec 18 2001
EXTENSIONS
a(8) from Giovanni Resta, Aug 21 2013
a(9)-a(11) from Farideh Firoozbakht confirmed and a(12) added by Max Alekseyev, Jul 30 2025
a(13) from Farideh Firoozbakht confirmed and added by Max Alekseyev, Jan 15 2026
STATUS
approved