A066229 f-perfect numbers, where f(m) = m + 1.

4, 10, 44, 2336, 8896, 34432, 449295, 549775212544, 2251801457852416, 9007202677293056, 9223372167851278336, 20055918935605248255

Offset: 1

Author

Joseph L. Pe, Dec 18 2001

Keywords

• nonn
• more

Comments

f-perfect numbers are defined in A066218.

From Farideh Firoozbakht, Sep 18 2006: (Start)

n is in the sequence iff sigma(n) = 2*n - d(n) + 2, where d(n) is number of positive divisors of n.

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.

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)

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

Examples

			f(10) = 11 = 2 + 3 + 6 = f(1) + f(2) + f(5), hence 10 is a term of the sequence.
		

Links

• Joseph L. Pe, On a Generalization of Perfect Numbers, J. Rec. Math., 31(3) (2002-2003), 168-172.
• Math StackExchange, Discussion on math.SE.

Crossrefs

Cf. A066218, A066230, A083874.

Programs

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[ # ] ] ] & ]

PARI

isok(m) = sigma(m) == 2*m-numdiv(m)+2; \\ Michel Marcus, Mar 13 2020

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