A066218 Numbers k such that sigma(k) = Sum_{j|k, j

198, 608, 11322, 20826, 56608, 3055150, 565344850, 579667086, 907521650, 8582999958, 13876688358, 19244570848, 195485816050, 255701999358, 1038635009650, 1410759512050, 3308222326688, 6293446033554, 12859914783762, 15343909268584, 18359652610976, 19142664182226, 41584649258178, 45090324794034, 56293124233554

Offset: 1

Author

Joseph L. Pe, Dec 17 2001

Keywords

• nonn
• more

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

No more terms < 10^14. - Jud McCranie, Nov 28 2024

Examples

			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).
		

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.

Crossrefs

Cf. A211779, A000203, A066229, A066230, A296075.

Programs

Mathematica

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

PARI

is(n)=sumdiv(n,d,sigma(d))==2*sigma(n) \\ Charles R Greathouse IV, Mar 09 2014

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

Extensions

More terms from Naohiro Nomoto, May 07 2002

a(7)-a(8) 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

a(17)-a(25) from Jud McCranie, Nov 28 2024