OFFSET
1,1
COMMENTS
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.
a(17) > 2*10^12. - Giovanni Resta, Jun 20 2013
Numbers k such that A296075(k) = 0. - Amiram Eldar, Apr 16 2024
LINKS
Joseph L. Pe, On a Generalization of Perfect Numbers, J. Rec. Math., 31(3) (2002-2003), 168-172.
Giovanni Resta, 34 numbers > 3*10^12 which belong to the sequence.
FORMULA
Integer n = p1^k1 * p2^k2 * ... * pm^km is in this sequence if and only if g(p1^k1)*g(p2^k2)*...*g(pm^km)=2, where g(p^k) = (p^(k+2)-(k+2)*p+k+1)/(p^(k+1)-1)/(p-1) for prime p and integer k>=1. - Max Alekseyev, Oct 23 2008
EXAMPLE
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).
MATHEMATICA
f[ x_ ] := DivisorSigma[ 1, x ]; Select[ Range[ 1, 10^5 ], 2 * f[ # ] == Apply[ Plus, Map[ f, Divisors[ # ] ] ] & ]
PROG
(PARI) is(n)=sumdiv(n, d, sigma(d))==2*sigma(n) \\ Charles R Greathouse IV, Mar 09 2014
CROSSREFS
KEYWORD
nonn,more
AUTHOR
Joseph L. Pe, Dec 17 2001
EXTENSIONS
More terms from Naohiro Nomoto, May 07 2002
2 more terms from Farideh Firoozbakht, Sep 18 2006
a(9)-a(13) from Donovan Johnson, Jun 25 2012
a(14)-a(16) from Giovanni Resta, Jun 20 2013
STATUS
approved