This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A066229 #52 Aug 01 2025 01:58:37
%S A066229 4,10,44,2336,8896,34432,449295,549775212544,2251801457852416,
%T A066229 9007202677293056,9223372167851278336,20055918935605248255
%N A066229 f-perfect numbers, where f(m) = m + 1.
%C A066229 f-perfect numbers are defined in A066218.
%C A066229 From _Farideh Firoozbakht_, Sep 18 2006: (Start)
%C A066229 n is in the sequence iff sigma(n) = 2*n - d(n) + 2, where d(n) is number of positive divisors of n.
%C A066229 If 2^(i+1) + 2*i - 1 is prime then n = 2^i*(2^(i+1) + 2*i - 1) is in the sequence because sigma(n) + d(n) - 2 = (2^(i+1) - 1)*(2^(i+1) + 2*i) + 2*(i+1) - 2 = 2^(2*i+2) + 2*i*2^(i+1) - 2^(i+1) = 2^(i+1)*(2^(i+1) + 2*i - 1) = 2*n, so sigma(n) = 2*n - d(n) + 2.
%C A066229 Hence if i is in {1, 2, 5, 6, 7, 19, 25, 26, 31, 38, 62, 80, 97, 110, 126, 133, 137, 409, 469, 685, 758, 1004, 1025, 1385, 2077, 2646, 2969, 3438, 7806, 8683, ...} then 2^i*(2^(i+1) + 2*i - 1) is in the sequence. 10, 44, 2336, 8896, 34432, 549775212544, 2251801457852416, 9007202677293056, 9223372167851278336, 151115727472444489859072, ... are such terms. (End)
%C A066229 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
%H A066229 Joseph L. Pe, <a href="https://vixra.org/abs/2503.0165">On a Generalization of Perfect Numbers</a>, J. Rec. Math., 31(3) (2002-2003), 168-172.
%H A066229 Math StackExchange, <a href="http://math.stackexchange.com/q/1834257#1834275">Discussion on math.SE</a>.
%e A066229 f(10) = 11 = 2 + 3 + 6 = f(1) + f(2) + f(5), hence 10 is a term of the sequence.
%t A066229 Select[ Range[ 500000 ], DivisorSigma[ 1, # ] == 2# - DivisorSigma[ 0, # ] + 2 & ] (* _Farideh Firoozbakht_, Sep 18 2006 *)
%t A066229 f[x_] := x + 1; Select[ Range[ 1, 10^5], 2 * f[ # ] == Apply[ Plus, Map[ f, Divisors[ # ] ] ] & ]
%o A066229 (PARI) isok(m) = sigma(m) == 2*m-numdiv(m)+2; \\ _Michel Marcus_, Mar 13 2020
%Y A066229 Cf. A066218, A066230, A083874.
%K A066229 nonn,more
%O A066229 1,1
%A A066229 _Joseph L. Pe_, Dec 18 2001
%E A066229 a(8) from _Giovanni Resta_, Aug 21 2013
%E A066229 a(9)-a(11) from _Farideh Firoozbakht_ confirmed and a(12) added by _Max Alekseyev_, Jul 30 2025