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 A066218 #51 Jul 30 2025 14:58:02
%S A066218 198,608,11322,20826,56608,3055150,565344850,579667086,907521650,
%T A066218 8582999958,13876688358,19244570848,195485816050,255701999358,
%U A066218 1038635009650,1410759512050,3308222326688,6293446033554,12859914783762,15343909268584,18359652610976,19142664182226,41584649258178,45090324794034,56293124233554
%N A066218 Numbers k such that sigma(k) = Sum_{j|k, j<k} sigma(j).
%C A066218 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).
%C A066218 Problems: Find an expression generating sigma-perfect numbers. Are there infinitely many of these? Find other interesting sets generated by other f's. 3.
%C A066218 a(17) > 2*10^12. - _Giovanni Resta_, Jun 20 2013
%C A066218 Numbers k such that A296075(k) = 0. - _Amiram Eldar_, Apr 16 2024
%C A066218 No more terms < 10^14. - _Jud McCranie_, Nov 28 2024
%H A066218 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 A066218 Giovanni Resta, <a href="/A066218/a066218.txt">34 numbers > 3*10^12 which belong to the sequence</a>.
%F A066218 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
%e A066218 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).
%t A066218 f[ x_ ] := DivisorSigma[ 1, x ]; Select[ Range[ 1, 10^5 ], 2 * f[ # ] == Apply[ Plus, Map[ f, Divisors[ # ] ] ] & ]
%o A066218 (PARI) is(n)=sumdiv(n,d,sigma(d))==2*sigma(n) \\ _Charles R Greathouse IV_, Mar 09 2014
%Y A066218 Cf. A211779, A000203, A066229, A066230, A296075.
%K A066218 nonn,more
%O A066218 1,1
%A A066218 _Joseph L. Pe_, Dec 17 2001
%E A066218 More terms from _Naohiro Nomoto_, May 07 2002
%E A066218 a(7)-a(8) from _Farideh Firoozbakht_, Sep 18 2006
%E A066218 a(9)-a(13) from _Donovan Johnson_, Jun 25 2012
%E A066218 a(14)-a(16) from _Giovanni Resta_, Jun 20 2013
%E A066218 a(17)-a(25) from _Jud McCranie_, Nov 28 2024