1,1

I propose this generalization of perfect numbers: for an arithmetical function f, the "f-perfect numbers" are the n such that f(n) = sum of f(k) where k ranges over proper divisors of n. The usual perfect numbers are i-perfect numbers, where i is the identity function. This sequence lists the sigma-perfect numbers. It is not hard to see that the EulerPhi-perfect numbers are the powers of 2, and the d-perfect numbers are the squares of primes (d(n) denotes the number of divisors of n).

Problems: Find an expression generating sigma-perfect numbers. Are there infinitely many of these? Find other interesting sets generated by other f's. 3.

J. Pe, On a Generalization of Perfect Numbers, J. Rec. Math., 31(3) (2002-2003), 168-172.

Proper divisors of 198 = {1, 2, 3, 6, 9, 11, 18, 22, 33, 66, 99}; sum of their sigma values = 1 + 3 + 4 + 12 + 13 + 12 + 39 + 36 + 48 + 144 + 156 = 468 = sigma(198).

f[ x_ ] := DivisorSigma[ 1, x ]; Select[ Range[ 1, 10^5 ], 2 * f[ # ] == Apply[ Plus, Map[ f, Divisors[ # ] ] ] & ]

Adjacent sequences: A066215 A066216 A066217 this_sequence A066219 A066220 A066221

Sequence in context: A055971 A075293 A083264 this_sequence A065697 A075457 A097731

nonn,more

Joseph L. Pe (joseph_l_pe(AT)hotmail.com), Dec 17 2001

More terms from Naohiro Nomoto (n_nomoto(AT)yabumi.com), May 07 2002

2 more terms from Farideh Firoozbakht (mymontain(AT)yahoo.com), Sep 18 2006